1. Problem and contribution

The task asks for imaginary quadratic fields $K=\mathbb Q(\sqrt{-D})$ with large $\ell$-rank $r_\ell=\dim_{\mathbb F_\ell}\mathrm{Cl}(K)/\ell$. Current searches [1,2] force two independent classes of order $\ell$ through simultaneous small representations

$$ 4m^\ell=y^2+Dz^2,\qquad m\ \text{small}, $$

and then pay one class-group computation per field for any further rank. A community cost model [3] puts the target $(11,4)$ at about $2\cdot10^8$ such computations. A cheap predictor of extra rank could therefore matter. For $\ell=3$ such a predictor exists. Llorente and Quer [4] used the connection between 3-ranks and ranks of Mordell curves with BSD-based rank estimates, as described in [9], and Elkies [8] applied the Mestre–Nagao heuristic [5,6] to Mordell curves to reach the record 3-rank 8. Bagshaw, Jacobson, Rollick and Scheidler list the extension to $p>3$ as open future work [1,9].

This paper is an empirical study of what such scores detect, for $\ell\in\{3,5,7,11\}$ ($\ell=13$ is not tested). It uses a main sample of 200,000 random fields with $D\in[10^9,10^{10}]$ and an independent replication sample of 199,979 fields. The findings, all within this range:

1. Mordell score. The score of $E_{-16D}:y^2=x^3-16D$ separates $r_3=1$ from $r_3=0$ (Welch $z=15.4$, effect $0.074$ SD). It enriches $r_3\ge2$ 27-fold among the top 0.1% of scores, and 23-fold in the replication.
2. Matched control without detected association. Quadratic twists of 27a are $E_{-16D^3}$, the cubic twist of $E_{-16D}$ by $D$. They draw on the same class-group input to the 3-isogeny descent (Section 3.5), yet no association of their score with $r_3$ is detected. The same holds for the quadratic twists of 19a, 11a, 26b and of a Jacobian isogenous to $J_0(23)$, scored with bad primes excluded. For $\ell=7$ and $11$ the rank-2 samples are small, so substantial effects cannot be excluded there; for example, the $\ell=11$ effect is $0.32\pm0.25$ SD.
3. The signal is concentrated in fields with rational points: small ones account for the mean and most of the tail, moderate-height ones for most of the rest. Integral points of $y^2=x^\ell-4^{\ell-1}Dz^2$ with $4\mid x$ correspond to representations $4m^\ell=y^2+Dz^2$, the classes pair searches force (Section 3.3). Fields with such a point score 1.2–1.7 SD higher. Imprimitive points, which carry no class-group content, raise the score just as much ($+1.0$ SD in fields with $r_3=0$). Fields with $r_3\ge1$ and no point show only small mean effects (at most $0.09$ SD). Small points account for 64–72% of the $r_3\ge2$ fields in the top 1% and top 0.1% of scores. Among fields without a point, however, the score still enriches $r_3\ge2$ ×4.3 (top 1%) and ×12.9 (top 0.1%). Representations with larger $m$ ($\le10^6$) or $2\le z\le13$ do not explain this residual: they are about equally common among tail and non-tail fields. A direct search for representations $4m^3=t^2+Dc^6$ with $c\le5$, each of which gives a rational point of $E_{-16D}$, finds one in 80% of the residual-tail fields with $r_3=2$, against 5% of low-scoring ones (Section 4.5). Each such point proves positive rank. This is consistent with the residual tail being largely the same point association at larger heights; it does not show that no other contribution remains.
4. Dose–response. Across cubic twists $E_{-16Dz^2}$, $z=1,2,3,5,7$, the $r_3=2$ effect falls from $0.32$ to $0.07$ SD while the per-point effect stays at 1.2–1.7 SD and the point frequency falls roughly like $z^{-1/3}$. No effect is detected for $z=D,D^2,2D$ ($|\Delta|\le0.021$ SD). Stratifying by $D\bmod8$ and $D\bmod243$ leaves the effects unchanged.
5. $\ell=5,7$. The same point effect appears on $y^2=x^\ell-4^{\ell-1}D$ ($+0.8$ to $+1.3$ SD with a point), but points are rare: 0.27% of fields with $r_5=1$ and 0.06% with $r_7=1$. In a further 1,000,000-field sample (2,049 fields with $r_5=2$, 493 with $r_7=2$) there is a weak residual tail for $\ell=5$ among fields without a point: ×1.5 in the top 1% and ×3.0 in the top 0.1%, about a third of the $\ell=3$ strength. These are two nested tests among eight fixed in advance, with nominal, unadjusted $p=0.024$ and $0.017$; neither is significant after a Bonferroni correction. This sample was a confirmatory test of a hypothesis raised by the main sample. Pooling with the main sample (exploratory) gives ×1.5 ($p=0.008$) and ×3.3 ($p=0.003$), but that mixes hypothesis generation with confirmation. A search for degree-2 divisors on the Jacobian, each a rank witness, finds them in 2 of 30 residual-tail fields against 3 of 200 low-scoring fields with $r_5=2$ (Section 4.6), so the source of the residual is still not identified. No residual is detected for $\ell=7$.

What follows, and what does not. For $\ell=3$, a direct search for small representations recovers the mean signal and about two-thirds of the score's top-tail hits, and it yields certificates. The score adds a substantial residual tail signal on top of that. For $\ell=5$ the score's information is mostly small points, which are rare as the heuristic frequency $D^{1/\ell-1/2}$ predicts, plus a weak residual tail. For $\ell=7$ only the rare small points are detected. We did not test scores inside pair-search pipelines, where every candidate already has two small representations. We also did not test other statistics (other cutoffs, multi-value or learned sums [19,20]) or larger $D$. So we make no claim about the cost of any target. The results identify what a Mestre–Nagao score measures and give a baseline that a pipeline-level benchmark would have to beat.

Class-group data are heuristic ground truth, validated as described in Section 3.1. No certificate for a target is claimed.

2. Related work and distinction

Rank heuristics and 3-ranks. Mestre [5] and Nagao [6] introduced partial sums $\sum_{p\le P}-a_p\log p/p$ as rank predictors. Bujanović, Kazalicki and Novak [19] show that Mestre–Nagao sums depend on the cutoff and the conductor range. Bujanović, Kazalicki and Vlah [20] compare multi-value and learned versions for rank classification. Neither addresses class-group prediction, but together they make clear that our results concern one statistic rather than every possible rank filter. Llorente and Quer [4] tabulated many fields with $r_3\ge4$; as described in [9], they used the 3-rank/Mordell-rank connection. Quer's three fields of 3-rank 6 came from Mordell curves of rank 12 [7,8]. Elkies [8] found $E_{16D}$ of rank 16 by Mestre–Nagao sieving and proved 3-rank 8 for $\mathbb Q(\sqrt{-3D})$. His 16 generators were found among integral points with $x\le10^{12}$.

Small representations. Diaz y Diaz [2] and BJRS [1] build fields from pairs of representations $4m_i^\ell=y_i^2+\lambda_i^2z^2D$. Our point counts use the same objects.

Descent and twists. Mestre [12] and Schoof [13] obtained 5-ranks from quadratic twists of 11a, and Gillibert–Levin [14] explain such constructions geometrically. For $y^2=x^\ell+A$, Stoll [10,11] relates Selmer ranks to class groups of $\mathbb Q(\zeta_\ell,\sqrt A)$. Average 3-isogeny Selmer sizes are treated in [15,16], and root numbers of $y^2=x^3+k$ in [17].

New here. We found no prior matched-field comparison of score families, no test for $\ell\ge5$, and no decomposition of a score's class-group signal by small representations. The yield model of [3] motivates the question and is not used in any argument.

3. Methods

3.1 Fields, ground truth and validation

Main sample: 200,000 distinct fundamental discriminants $-D$ with $D$ uniform in $[10^9,10^{10}]$ (seed 11). Replication sample: 199,979 fields (seed 99), with the 21 overlaps removed.

The class number $h$ is computed from an Euler-product estimate of $L(1,\chi_{-D})$ ($p\le10^5$) followed by baby-step giant-step on prime forms. $r_\ell$ is read off the enumerated $\ell$-Sylow subgroup. Validation:

• Small discriminants. Exhaustive enumeration of reduced forms for 300 fields with $D<3\cdot10^6$ (600 checks of $(h,r_3)$ and $(h,r_5)$): 0 mismatches.
• Sampled range. Recomputation with $p\le10^6$ for 999 fields of the main sample: all 94 high-score $r_3=2$ fields of Section 4.3, 306 further $r_3=2$ fields (including non-cyclic 3-Sylow subgroups such as $C_{3^5}\times C_3$) and 599 random fields. $h$ and $(r_3,r_5,r_7,r_{11})$ agreed in 999 of 999 cases.
• Exact lower bounds. For 200 fields with heuristic $r_3=2$, including all 94 high-score ones, two independent forms of order 3 were found and verified by exact composition and reduction (forms.py), so $r_3\ge2$ is proved unconditionally. Upper bounds are not certified.
• Cross-check against points. No field with heuristic $r_3=0$ has a primitive representation with $z\in\{1,2\}$, although every such representation forces a 3-torsion class.

Rank counts in the main sample: $r_3=0,1,2,3$: 113,380 / 83,111 / 3,499 / 10. $r_5$: 151,805 / 47,803 / 392 / 0. $r_7$: 167,279 / 32,631 / 90 / 0. $r_{11}$: 180,188 / 19,795 / 17 / 0.

3.2 Scores

$$ S_P(A)=\sum_{p\le P,\ p\nmid N_A}-\frac{a_p(A)\log p}{p},\qquad P=2\cdot10^5 . $$

The families:

• Mordell family $E_k:y^2=x^3+k$, $k=-16Dz^2$. Only $q\equiv1\pmod3$ contribute, with $a_q$ determined by $k\in\mathbb F_q^\times/\mathbb F_q^{\times6}$. Primes dividing $6D$ are excluded.
• CM family $y^2=x^\ell+h$, $h\in\{-D,-4^{\ell-1}D\}$. Only $q\equiv1\pmod\ell$ contribute, via $h\in\mathbb F_q^\times/\mathbb F_q^{\times2\ell}$. Primes dividing $2\ell D$ are excluded.
• Quadratic twists $a_p(A^{(-D)})=\chi_{-D}(p)a_p(A)$, for 19a3, 27a3 ($\ell=3$), 11a3 ($\ell=5$), 26b1 ($\ell=7$) and the genus-2 curve LMFDB 529.a.529.1 [18], $y^2=x^6-4x^5+2x^4+2x^3+x^2+2x+1$, whose Jacobian is isogenous to $J_0(23)$ with $J(\mathbb Q)\cong\mathbb Z/11$ per LMFDB. Primes dividing the conductor of $A$ (and 2 for the genus-2 model) are excluded. Primes dividing $D$ contribute 0. Rational $\ell$-torsion was checked for the elliptic curves ($\ell\mid\#E(\mathbb F_p)$ for all good $p<2\cdot10^5$). For the genus-2 curve we matched $\mathrm{tr}\,a_{3,5,7}=(0,-2,2)$ with the level-23 newform.

All traces were cross-checked against point counts. The Mordell scorer at $z=1$ and $z=D$ reproduces, to four decimals, independent implementations for $h=-16D$ and for the 27a twist, confirming $27\mathrm a^{(-D)}\cong E_{-16D^3}$.

3.3 Small representations and small points

With $X=4m$ and $Y=2^{\ell-1}y$,

$$ 4m^\ell=y^2+Dz^2\iff Y^2=X^\ell-4^{\ell-1}Dz^2 . $$

For $\ell=3$ the right-hand side is $E_{-16Dz^2}$. Also $E_{-D}$ corresponds to $z=2$, since $y^2=x^3-D\iff(2y)^2+4D=4x^3$.

Each representation gives $\alpha=(y+z\sqrt{-D})/2\in\mathcal O_K$ of norm $m^\ell$. It is primitive if no rational prime divides $\alpha$. That fails for an odd $p$ when $p\mid\gcd(y,z)$. At $p=2$ it fails when $y,z$ are even and either $D\equiv3\pmod 4$ with $y/2\equiv z/2\pmod2$, or $4\mid D$ with $4\mid y$. A primitive representation with $1<m<D/4$ forces a non-principal class of order $\ell$, and a $z=1$ representation is always primitive.

We enumerate all representations with $m\le M$, classifying each one: $M=3\cdot10^4$ ($\ell=3$), $4\cdot10^3$ ($\ell=5$) and $10^3$ ($\ell=7$). "Extended" representations in Section 4.3 are $z=1$ with $m\le10^6$, or $2\le z\le13$ with $m\le3\cdot10^4$.

3.4 Statistics

Effect sizes are Glass's $\Delta$: $(\bar S_1-\bar S_0)/s_0$, with $s_0$ the SD of the reference group ($r=0$, or $r=0$ without a point where stated). Standard errors are Welch's, $\sqrt{s_1^2/n_1+s_0^2/n_0}/s_0$. z-scores are Welch statistics. Stratum-adjusted effects combine per-stratum differences with sample-size weights $n_1n_0/(n_1+n_0)$. Tails are measured in two ways: by exceedance of the overall top-1% score threshold, relative to the reference group, and by top-$x\%$ enrichment $P(r\ge2\mid\text{top }x\%)/P(r\ge2)$ with one-sided Poisson $p$-values. As a consistency check, every quadratic-twist score shifts by 0.3–0.7 with the root-number sign $\chi_{-D}(N)$, as expected of a statistic that responds to analytic rank.

3.5 Selmer structure of the matched pair

For $k=-16Dz^2$, the 3-isogeny $E_k\to E_{-27k}$ has kernel $\mathbb Z/3(\chi_{-D})$ and dual kernel $\mathbb Z/3(\chi_{3D})$ for every $z$. The unramified parts of the two $H^1$'s are $\mathrm{Hom}(\mathrm{Cl}(\mathbb Q(\sqrt{-D})),\mathbb Z/3)$ and $\mathrm{Hom}(\mathrm{Cl}(\mathbb Q(\sqrt{3D})),\mathbb Z/3)$. At $p\ge5$, $p\mid D$, the characters are ramified, so $H^0=H^2=0$ and therefore $H^1(\mathbb Q_p,\cdot)=0$. Hence the Selmer groups of $E_{-16D}$ and $E_{-16D^3}$ are cut out of the same ambient group of classes unramified outside $\{2,3\}$, defined by the same Galois modules and class-group data, with possibly different local conditions at 2 and 3. This does not identify the actual Selmer groups, their Mordell–Weil images or their Tate–Shafarevich contributions. It shows only that the two curves draw on the same class-group input, so the contrast between their scores is not explained by different class groups entering the descent.

4. Results

4.1 All families

Table 1. All families on the main sample ($P=2\cdot10^5$, bad primes excluded). z: Welch statistics against $r=0$. Effects: Glass's $\Delta$ with Welch SE. Enrichment: $P(r\ge2\mid\text{top }x\%)/P(r\ge2)$, with hits and one-sided Poisson $p$. Rank-2 counts are 3,499 ($\ell=3$), 392 (5), 90 (7) and 17 (11).

| family, $P=2\cdot10^5$ | $\ell$ | Welch z, $r=1$ | Welch z, $r=2$ | effect $r=1$ | effect $r=2$ | top 1%: enrichment (hits; $p$) | top 0.1% | |---|---|---|---|---|---|---|---| | Mordell $E_{-16D}$ | 3 | +15.4 | +13.6 | $+0.074\pm0.005$ | $+0.312\pm0.023$ | ×8.3 (291; $p<10^{-15}$) | ×27.4 (96; $p<10^{-15}$) | | Mordell $E_{-D}$ | 3 | +9.1 | +7.7 | $+0.043\pm0.005$ | $+0.163\pm0.021$ | ×5.3 (187; $p<10^{-15}$) | ×15.7 (55; $p<10^{-15}$) | | quadratic twists of 27a ($=E_{-16D^3}$) | 3 | +0.7 | -0.6 | $+0.003\pm0.005$ | $-0.011\pm0.017$ | ×1.0 (35; p=0.53) | ×0.6 (2; p=0.87) | | quadratic twists of 19a | 3 | +0.8 | +1.2 | $+0.003\pm0.005$ | $+0.020\pm0.017$ | ×0.9 (32; p=0.72) | ×1.1 (4; p=0.47) | | CM $y^2=x^5-D$ | 5 | +1.0 | +0.4 | $+0.005\pm0.005$ | $+0.023\pm0.054$ | ×1.5 (6; p=0.2) | ×5.1 (2; p=0.059) | | CM $y^2=x^5-4^4D$ | 5 | +2.2 | +0.1 | $+0.012\pm0.005$ | $+0.004\pm0.056$ | ×2.6 (10; p=0.0071) | ×10.2 (4; p=0.00072) | | quadratic twists of 11a | 5 | -1.6 | +0.2 | $-0.008\pm0.005$ | $+0.009\pm0.051$ | ×0.8 (3; p=0.75) | ×0.0 (0; p=1) | | CM $y^2=x^7-D$ | 7 | -1.7 | -1.3 | $-0.010\pm0.006$ | $-0.144\pm0.113$ | ×4.4 (4; p=0.013) | ×0.0 (0; p=1) | | CM $y^2=x^7-4^6D$ | 7 | -0.2 | +0.9 | $-0.001\pm0.006$ | $+0.105\pm0.119$ | ×2.2 (2; p=0.23) | ×11.1 (1; p=0.086) | | quadratic twists of 26b | 7 | -0.5 | +1.0 | $-0.003\pm0.006$ | $+0.100\pm0.098$ | ×0.0 (0; p=1) | ×0.0 (0; p=1) | | quadratic twists of 529.a.529.1 ($\sim J_0(23)$) | 11 | -0.2 | +1.2 | $-0.001\pm0.008$ | $+0.316\pm0.254$ | ×0.0 (0; p=1) | ×0.0 (0; p=1) |

Replication (199,979 fields): Mordell $E_{-16D}$ has Welch z $+14.4$ ($r_3=1$) and $+12.4$ ($r_3=2$), with enrichment ×8.3 (300 hits) and ×23.0 (83 hits). For $y^2=x^5-4^4D$: ×1.6 (7 hits, $p=0.14$) and ×4.7 (2 hits, $p=0.07$). As in the main sample, no replication field with $r_3=0$ has a $z=1$ point.

4.2 Decomposition by small points

Table 2. Effect sizes against fields with $r=0$ and no point (Glass's $\Delta$, Welch SE). "Tail" is the rate of exceeding the overall top-1% score threshold, relative to that reference group. Points: representations with $m\le M$ as in Section 3.3. For $E_{-D}$ only fields whose $z=2$ representations are all primitive count as "with point". Primitivity was classified for every representation, not only the smallest.

| curve | $\ell$ | point fields $r=1$ / $r=2$ | $r=1$ with point | $r=1$ without | $r=2$ with point | $r=2$ without | |---|---|---|---|---|---|---| | $E_{-16D}$, $z=1$, $m\le3\cdot10^4$ | 3 | 3818 / 572 | $+1.15\pm0.02$; tail ×17.4 | $+0.022\pm0.005$; tail ×2.4 | $+1.68\pm0.06$; tail ×68.7 | $+0.045\pm0.022$; tail ×7.7 | | $E_{-D}$, primitive $z=2$ | 3 | 1656 / 216 | $+1.17\pm0.02$; tail ×17.1 | $+0.020\pm0.005$; tail ×2.1 | $+1.26\pm0.08$; tail ×25.8 | $+0.091\pm0.021$; tail ×8.1 | | $y^2=x^5-4^4D$, $z=1$, $m\le4\cdot10^3$ | 5 | 129 / 7 | $+1.27\pm0.10$; tail ×19.6 | $+0.008\pm0.005$; tail ×1.0 | $+2.47\pm0.50$; tail ×43.3 | $-0.041\pm0.054$; tail ×1.8 | | $y^2=x^7-4^6D$, $z=1$, $m\le10^3$ | 7 | 20 / 1 | $+0.84\pm0.23$; tail ×10.0 | $-0.002\pm0.006$; tail ×1.0 | – | $+0.067\pm0.114$; tail ×1.1 |

Imprimitive control. For $E_{-D}$, fields whose $z=2$ representations are all imprimitive, and so carry no class-group content, score $+1.02\pm0.02$ SD ($r_3=0$, $n=1606$), $+1.20\pm0.02$ ($r_3=1$, $n=2071$) and $+1.38\pm0.11$ ($r_3=2$, $n=165$). Only 92 fields have both kinds. Replication, $\ell=5$: $+1.38\pm0.09$ SD for the 126 fields with $r_5=1$ and a point, and $+0.000\pm0.005$ for the 47,589 without.

4.3 The residual tail

Table 4. Upper tails of the $E_{-16D}$ score. Fields with $r_3=2$ are all those in the main sample. Fields with $r_3\le1$ are a control set of 20,000 drawn with shuf using a fixed byte stream as its randomness source. That is deterministic but not a seeded pseudorandom sampler, so we ship the exact list (data/ctrl20k.txt) instead of claiming uniformity. The residual-tail enrichments below use the full no-point samples and do not depend on this control set. "Extended" = a representation with $z=1$, $m\le10^6$, or $2\le z\le13$, $m\le3\cdot10^4$ (primitive or not).

| group | $n$ | share with extended rep. | $P(S>20)$ [95% CI] | $P(S>15)$ | |---|---|---|---|---| | $r_3=0$ | 11,630 | 0.135 | 0.0028 [0.0019, 0.0039] | 0.024 | | $r_3=1$ | 8,370 | 0.370 | 0.0133 [0.0110, 0.0159] | 0.047 | | $r_3=2$, no $z=1$ point | 2,927 | 0.687 | 0.0321 [0.0263, 0.0391] | 0.065 | | — with an extended rep. | 2,010 | 1 | 0.0343 [0.0272, 0.0432] | 0.070 | | — with no extended rep. | 917 | 0 | 0.0273 [0.0185, 0.0399] | 0.055 | | $r_3\le1$, no extended rep. | 15,331 | 0 | 0.0046 [0.0037, 0.0058] | 0.028 |

Extended representations are about as common among high-score fields as among all fields with $r_3=2$: 69 of the 94 fields with $S>20$, against a share of 0.687. Fields with $r_3=2$ and none of them still exceed $S>20$ six times as often as comparable fields with $r_3\le1$. Restricting to the 195,606 main-sample fields without a $z=1$ point, the top 1% and top 0.1% of scores enrich $r_3\ge2$ ×4.3 (125 hits against 29.3 expected) and ×12.9 (38 against 2.9), $p<10^{-15}$. For $y^2=x^5-4^4D$ the same restriction gives ×1.8 (7 against 3.9, $p=0.10$) and ×5.2 (2 against 0.4, $p=0.06$). Among $r_3\ge2$ fields in the top 1% and top 0.1% of the $E_{-16D}$ score, 64% and 72% have a $z=1$ point. Replication: among the 195,640 replication fields without a $z=1$ point, the same enrichment is ×4.4 (133 against 30.2 expected) and ×10.9 (33 against 3.0), $p<10^{-15}$. There, 61% and 75% of the top-tail fields with $r_3\ge2$ have a $z=1$ point.

4.4 Cubic twists, $D$-dependence and local data

Table 3. Cubic twists $E_{-16Dz^2}$ on the probe set: the 100,000 smallest $D$ plus all 3,499 fields with $r_3=2$ ($n=101{,}749$; $r_3=2$ oversampled, so effects are against the probe set's own $r=0$). Point frequency is for $4m^3=y^2+Dz^2$, $m\le3\cdot10^4$, all representations counted. For $z\in\{D,D^2,2D\}$ no representation has $m\le3\cdot10^4$.

| $z$ | effect $r_3=1$ | effect $r_3=2$ | point freq. $r_3=1$ / $r_3=2$ | $r_3=2$ with point | $r_3=2$ without | |---|---|---|---|---|---| | $1$ | $+0.077\pm0.007$ | $+0.319\pm0.023$ | 0.050 / 0.163 | $+1.69\pm0.06$ | $+0.051\pm0.022$ | | $2$ | $+0.039\pm0.007$ | $+0.164\pm0.021$ | 0.049 / 0.119 | $+1.43\pm0.06$ | $-0.007\pm0.021$ | | $3$ | $+0.037\pm0.007$ | $+0.122\pm0.021$ | 0.044 / 0.102 | $+1.49\pm0.07$ | $-0.034\pm0.020$ | | $5$ | $+0.031\pm0.007$ | $+0.106\pm0.020$ | 0.034 / 0.092 | $+1.41\pm0.08$ | $-0.026\pm0.020$ | | $7$ | $+0.018\pm0.007$ | $+0.070\pm0.020$ | 0.031 / 0.080 | $+1.24\pm0.07$ | $-0.032\pm0.019$ | | $D$ | $+0.000\pm0.006$ | $-0.014\pm0.018$ | | $D^2$ | $-0.012\pm0.006$ | $-0.021\pm0.018$ | | $2D$ | $+0.013\pm0.006$ | $-0.004\pm0.017$ |

Dependence on $D$ ($E_{-16D}$, quartiles of $D$, medians $2.1$ to $8.9\cdot10^9$): effect for $r_3=1$ $0.082, 0.072, 0.071, 0.070$ ($\pm0.010$); for $r_3=2$ $0.408, 0.312, 0.310, 0.219$ ($\pm0.045$). The $z=1$ point frequency falls from 0.191 to 0.148 ($r_3=2$) and from 0.055 to 0.042 ($r_3=1$). Both ratios, 0.77 and 0.76, are close to $(8.9/2.1)^{-1/6}=0.79$. With a fixed cutoff $M$ the agreement is approximate.

Local data. The stratum-adjusted effect of $E_{-16D}$ equals the raw effect, both within the 96 classes $(D\bmod8,D\bmod27)$ and within the 864 classes $(D\bmod8,D\bmod243)$: $0.0736$, $0.0736$ and $0.0737$ ($\pm0.0048$) for $r_3=1$, and $0.312$, $0.312$ and $0.311$ ($\pm0.023$) for $r_3=2$. The product of Liverance's local root-number factors at $p\mid D$ is constant within these classes, and $r_3$ does not depend on it ($P(r_3\ge1)=0.4334$ against $0.4328$). So local data at 2 and 3, including the root-number factors, account for none of the signal.

4.5 Most residual-tail fields have rational points of moderate height

Section 4.3 leaves open where the residual tail comes from. A representation $4m^3=t^2+Dc^6$, i.e. $z=c^3$, gives the rational point $(x,y)=(4m/c^2,\,4t/c^3)$ of $E_{-16D}$. The converse with the same $c$ needs divisibility conditions on the numerators, so not every rational point with denominator parameter $c$ arises this way; we therefore describe the search by the representations searched. We searched for representations with $c=1$ ($m\le10^6$), $c=2$ ($m\le4\cdot10^6$), $c=3$ ($m\le9\cdot10^6$), $c=4$ ($m\le1.6\cdot10^7$) and $c=5$ ($m\le2.5\cdot10^7$) in four groups of main-sample fields, all without a $z=1$ point with $m\le3\cdot10^4$. Positive results are rank witnesses; negative results exclude only these representations, not all points of small height. For these $D$, $k=-16D$ is neither a square, a cube nor $-432$, so $E_{-16D}(\mathbb Q)_{\mathrm{tors}}=0$ and every point found proves $\operatorname{rank}E_{-16D}(\mathbb Q)\ge1$.

Table 5. Fields with a representation $4m^3=t^2+Dc^6$, $c\le5$, in the stated ranges (each gives a rational point of $E_{-16D}$). Wilson 95% intervals. Groups B, C1 and C0 are seeded random draws (seed 11) from the stated sets.

| group | $n$ | with a point | share [95% CI] | found at $c=1,2,3,4,5$ | |---|---|---|---|---| | A: $r_3=2$, $S>20$ (the residual-tail fields) | 94 | 75 | 0.80 [0.71, 0.87] | 19, 48, 37, 58, 33 | | B: $r_3=2$, $S$ below the median | 200 | 10 | 0.05 [0.03, 0.09] | 2, 6, 2, 8, 4 | | C1: $r_3=1$, $S>20$ | 150 | 54 | 0.36 [0.29, 0.44] | 7, 27, 18, 30, 22 | | C0: $r_3=0$, $S>20$ | 150 | 0 | 0.00 [0.00, 0.02] | 0, 0, 0, 0, 0 |

So 80% of the residual-tail fields with $r_3=2$ provably have positive rank through a point of moderate height, against 5% of low-scoring fields with $r_3=2$. The high scores of fields with $r_3=0$ come with no point in this range. That is consistent with their exceedance being chance fluctuation, or rank from points beyond the searched range; we cannot distinguish the two. These observations associate high scores with discoverable positive rank. They are consistent with the residual tail of Section 4.3 being largely the same point association at heights beyond the bounded small representations of Table 4. They do not measure how much class-group enrichment remains after conditioning on these points, and they do not exclude another contribution to the residual.

4.6 A larger test for $\ell=5,7$

To test for a residual signal at $\ell=5,7$ with more power, we drew 1,000,000 further fields with $D\in[10^9,10^{10}]$ using a seeded sampler (seeded_sample.py, seed 2026), excluding the main and replication samples. Rank counts: $r_5=0,1,2$: 761,674 / 236,277 / 2,049. $r_7=0,1,2$: 837,581 / 161,926 / 493. $r_3=0,1,2,3$: 567,673 / 414,684 / 17,585 / 58. Effects are against fields with $r=0$ and no $z=1$ point.

Table 6. The 1,000,000-field sample. Point = a $z=1$ representation ($m\le3\cdot10^4$ for $\ell=3$, $4\cdot10^3$ for $\ell=5$, $10^3$ for $\ell=7$). Residual enrichment = $P(r\ge2\mid\text{top }x\%)/P(r\ge2)$ among the fields without a point.

| curve | point fields $r=1$ / $r=2$ | $r=1$, with point | $r=1$, without | $r=2$, with point | $r=2$, without | residual enrichment, top 1% / 0.1% | |---|---|---|---|---|---|---| | Mordell $E_{-16D}$ ($\ell=3$, $m\le3\cdot10^4$) | 18,759 / 2,957 | $+1.14\pm0.01$ | $+0.014\pm0.002$ | $+1.63\pm0.03$ | $+0.018\pm0.010$ | ×4.6 (674 vs 146.6, $p<10^{-15}$) / ×13.0 (191 vs 14.7, $p<10^{-15}$) | | $y^2=x^5-4^4D$ | 650 / 35 | $+1.09\pm0.04$ | $-0.002\pm0.002$ | $+1.12\pm0.20$ | $+0.029\pm0.023$ | ×1.49 (30 vs 20.1, $p=0.024$) / ×3.0 (6 vs 2.0, $p=0.017$) | | $y^2=x^5-D$ | 650 / 35 | $-0.03\pm0.04$ | $-0.001\pm0.002$ | $-0.16\pm0.18$ | $+0.028\pm0.022$ | ×0.8 (16 vs 20.1) / ×0.5 (1 vs 2.0) | | $y^2=x^7-4^6D$ | 112 / 5 | $+1.06\pm0.09$ | $-0.002\pm0.003$ | $+0.51\pm0.17$ | $+0.026\pm0.046$ | ×0.4 (2 vs 4.9) / ×0 (0 vs 0.5) | | $y^2=x^7-D$ | 112 / 5 | $-0.08\pm0.09$ | $-0.005\pm0.003$ | $-0.33\pm0.56$ | $+0.057\pm0.044$ | ×1.0 (5 vs 4.9) / ×0 (0 vs 0.5) |

For $\ell=3$ the 1,000,000-field sample is a third independent replication. Over all fields the Mordell score enriches $r_3\ge2$ ×8.7 in the top 1% (1,537 hits) and ×27.5 in the top 0.1% (485 hits). $z=1$ points occur in 4.5% of fields with $r_3=1$ and 16.8% with $r_3=2$, and none with $r_3=0$. Among fields without a point the residual tail is ×4.6 and ×13.0.

For $\ell=5$ on the curve $y^2=x^5-4^4D$ the confirmatory sample shows a weak residual tail. The analysis script (big_an.py) was written before this sample had been scored, and it fixed the comparisons in advance: the residual enrichment of $r\ge2$ among fields without a point, at the top 1% and the top 0.1% of each score in Table 6, the statistics of Section 4.3. For $\ell=5,7$ that is eight one-sided residual tests; the script also printed the all-field enrichments for the same curves. On $y^2=x^5-4^4D$ they give ×1.49 (30 against 20.1 expected) and ×3.0 (6 against 2.0), with nominal, unadjusted $p=0.024$ and $p=0.017$. The two are nested, since the top 0.1% lies inside the top 1%, so they are not independent confirmations, and neither survives a Bonferroni correction over the eight tests ($0.05/8\approx0.006$). The other six give no excess. The population thresholds and aggregate counts are in data/big/tails.tsv. Exploratory pooling with the hypothesis-generating main sample, which overstates the evidence, gives ×1.5 (37 against 24.0, $p=0.008$) and ×3.3 (8 against 2.4, $p=0.003$). Fields with $r_5=1$ and no point show no tail excess (exceedance ratio 1.03). For $\ell=7$ no residual is detected (493 fields with $r_7=2$).

Points and divisors for $\ell=5$. Let $C_D:y^2=f(x)=x^5-4^4D$ and $J_D$ its Jacobian. We tested four groups from the 1,000,000-field sample, all without a $z=1$ point ($m\le4\cdot10^3$): A5, the 30 fields with $r_5\ge2$ in the top 1% of scores (the residual-tail fields); B5, 200 fields with $r_5=2$ and score below the median; C15 and C05, 150 each with $r_5=1$ and $r_5=0$ in the top 1% (seeded draws).

Curve points. A representation $4m^5=t^2+Dc^{10}$ gives the point $(4m/c^2,\,16t/c^5)$ of $C_D$. We searched $c=1$ ($m\le10^6$), $c=2$ ($m\le3\cdot10^6$) and $c=3$ ($m\le6\cdot10^6$) in A5 and B5 and found none. This excludes only these representations. It says nothing about $\operatorname{rank}J_D(\mathbb Q)$, because a point of $J_D(\mathbb Q)$ need not be of the form $P-\infty$ with $P\in C_D(\mathbb Q)$.

Degree-2 divisors. Every point of $J_D(\mathbb Q)$ is $E-\deg(E)\infty$ for a unique reduced effective divisor $E$ of degree $\le2$, and those of degree 2 have Mumford coordinates $(x^2+ux+v,\ ax+b)$ with $x^2+ux+v\mid f-(ax+b)^2$. Writing $x^5\equiv\alpha x+\beta$ modulo $x^2+ux+v$, with $\alpha=u^4-3u^2v+v^2$ and $\beta=uv(u^2-2v)$, the condition is that $s=a^2$ solves $(u^2-4v)s^2+(2\alpha u-4(\beta+h))s+\alpha^2=0$, $h=-4^4D$, with $s$ a nonzero rational square. We searched all $u,v$ in the three boxes of Table 7, two integral and one with denominator parameter 2 (divsearch2.c, with a quadratic-residue sieve), and rechecked every hit with exact rational arithmetic (dv_verify.py). The divisors $(q,L)$ and their classes are listed in data/dv/witnesses.txt. For these $D$, $J_D(\mathbb Q)_{\mathrm{tors}}=0$. $f$ is irreducible, so there is no rational 2-torsion. For good $p\equiv2,3\pmod5$, $\#J_D(\mathbb F_p)=p^2+1$, and the gcd over two such primes not dividing $10D$ leaves only 2 and 5 (checked for every $D$ with primes $\le97$ in dv_verify.py). A rational point of order 5 would generate a finite $\mathbb Z[\zeta_5]$-module. Since $\mathbb Z[\zeta_5]/(5)$ is local, that module has a nonzero element killed by $1-\zeta_5$, i.e. a multiple of $T=(0,\sqrt h)-\infty$, which generates $J_D[1-\zeta_5]$. That element is fixed by $\mathrm{Gal}(\bar{\mathbb Q}/\mathbb Q(\zeta_5))$, so $T$ would be defined over $\mathbb Q(\zeta_5)$, forcing $\sqrt{-D}\in\mathbb Q(\zeta_5)$. So every hit proves $\operatorname{rank}J_D(\mathbb Q)\ge1$. Under $y-\sqrt h$, each divisor maps to an element $\gamma\in K$ of norm $v^5$, and we computed the class of the ideal $\mathfrak a$ with $\mathfrak a^5=(\gamma)$ (dv_classes.py).

Table 7. Fields with a degree-2 divisor on $C_D$ in the stated boxes. Wilson 95% intervals.

| group | $n$ | fields with a divisor | share [95% CI] | |---|---|---|---| | A5: $r_5\ge2$, top 1%, no point (the residual-tail fields) | 30 | 2 | 0.07 [0.02, 0.21] | | B5: $r_5=2$, score below the median | 200 | 3 | 0.015 [0.005, 0.043] | | C15: $r_5=1$, top 1% | 150 | 0 | 0.00 [0.00, 0.025] | | C05: $r_5=0$, top 1% | 150 | 0 | 0.00 [0.00, 0.025] |

Boxes: (1) integral $|u|\le500$, $|v|\le10^6$; (2) integral $|u|\le1500$, $|v|\le5\cdot10^6$; (3) denominator parameter 2, $u=U/4$, $v=V/16$ with $|U|\le1000$, $|V|\le4\cdot10^6$, searched as integral $U,V$ on $y^2=x^5-2^{18}D$. Every divisor found lies in box 1; boxes 2 and 3 add none (box 3 finds only the box-1 divisors rescaled).

All six divisors (in five fields) have $|u|\le392$ and $v\le1.2\cdot10^5$, so their roots have absolute value close to $|h|^{1/5}$, the natural scale of the curve. Each maps under $y-\sqrt h$ to a class of exact order 5; the two in one field give mutually inverse classes. So divisors of this height occur in a few fields with $r_5=2$ and in none of the 300 high-scoring fields with $r_5\le1$. A divisor in a field with $r_5=0$ would need a trivial image, and none of those found has one. The residual-tail rate exceeds the low-score rate (one-sided Fisher $p=0.13$), but not significantly. Only about 10 of the 30 A5 fields are in excess of the 20.1 expected, so only a large effect could have been seen. Unlike the $\ell=3$ points of Section 4.5, divisors in this range do not account for the $\ell=5$ residual, and its source remains unidentified. They do give exact witnesses that $J_D(\mathbb Q)$ has positive rank with nonzero image in $\mathrm{Cl}(K)[5]$ for these five fields (data/dv/).

5. Interpretation

Small points and the residual. A field gains analytic rank on $E_{-16Dz^2}$ through points of small height, i.e. small $z$-representations, or through points of larger height. The imprimitive control shows that the score responds to points as such, and correlates with $r_3$ because primitive small points are 3-torsion classes. The mean effect is carried mostly by $z=1$ points with $m\le3\cdot10^4$. In the residual tail, Section 4.5 finds representations with $c\le5$, hence rational points, in 80% of the fields with $r_3=2$, proving positive rank; low-scoring fields with $r_3=2$ have them in only 5%. In our data the association between the score and $r_3$ is therefore concentrated in fields with rational points of $E_{-16D}$ across a range of heights. Small points are the most frequent, but not the only, case. This remains an association, not a causal account. In practice a direct search for small representations recovers most of the signal, and the score adds the fields whose points are too high for a bounded search.

Frequency heuristic. For random $D$ and fixed $z$, the number of pairs with $4m^\ell-y^2=Dz^2$ is about $(Dz^2)^{1/\ell-1/2}$ up to local density. For $\ell=3$ this gives $z^{-1/3}$ and $D^{-1/6}$, both consistent with Section 4.4; with a fixed cutoff $M$ the agreement is only approximate. For $\ell=5$ and $7$ the frequencies are 10–100 times smaller at $D\approx5\cdot10^9$, as observed.

Quadratic twists. For $A$ with rational $\ell$-torsion, points on $A^{(-D)}$ are not small representations. For 27a they correspond to $z=D$. No association of these scores with $r_\ell$ is detected in our data, although their descents draw on the same class-group input. The sizes of effect our samples can exclude are given in Table 1. Mestre's and Schoof's use of the implication from high-rank twists to large class-group rank is unaffected.

6. Limitations

• Ground truth is heuristic. It was validated by recomputation (999/999) and exact lower-bound certification for 200 fields with $r_3=2$, but not certified in general. Classification errors that depend on arithmetic could bias associations, which is why the tail fields were validated specifically.
• Only $D\in[10^9,10^{10}]$, $P=2\cdot10^5$ and the listed families are covered. Other statistics [19,20] and larger $D$ were not tested.
• Pipeline-conditioned candidates, with two forced classes, and the comparison of scoring, point-search and class-group costs were not tested. No claim about target search costs is made.
• In the main sample there are 90 fields with $r_7=2$ and only 17 with $r_{11}=2$; the 1,000,000-field sample adds 2,049 with $r_5=2$ and 493 with $r_7=2$, but $\ell=11$ was not rescored there and $\ell=13$ was not tested. The $\ell=5$ residual is weak: nominal, unadjusted $p=0.024$ and $0.017$ for two nested tests in the confirmatory sample, not significant after correcting for the eight tests fixed in advance. The pooled exploratory values ($0.008$, $0.003$) mix hypothesis generation with confirmation.
• Representation counts are bounded ($m\le M$, $z\le13$), and Mordell–Weil ranks were bounded below only by points from representations with $c\le5$ in the stated ranges. For $\ell=5$, Jacobian points were searched only as degree-2 divisors with integral or denominator-2 Mumford coordinates in the boxes of Table 7. About 20% of the residual-tail fields with $r_3=2$ have no point in that range, and their source is not identified.

7. Reproducibility

run_all.sh in the supplement regenerates every data file of Sections 4.1–4.4 from the seeds: sampling, overlap removal, sharding, trace tables (with bad primes removed), ranks, scores, representation counts, the probe and tail subsets, and the replication. The 20,000-field control list of Table 4 is shipped and reused. It then runs the analysis scripts (stats2.py, tables_rev.py, strat2.py, replicate.py, validate_cg.py, certify_r2.py, hp_test.py), which print the numbers in Sections 4.1–4.5. This takes about 6 hours on 4 cores. run_big.sh regenerates the 1,000,000-field sample of Section 4.6 with a seeded sampler (seeded_sample.py), big_an.py prints its numbers, big_tails.py its population thresholds and counts (shipped as data/big/tails.tsv), and hp_test.py l5 prints the $\ell=5$ point test. It then runs the divisor search of Table 7 (divsearch2.c, dv_test.py, dv_verify.py, dv_classes.py; about 1.5 hours on 4 cores for the three boxes), about 3 hours in all. The 2 MB supplement limit rules out the full tables. The data files instead contain every main-sample field with $r_3\ge2$, a random subsample of 5,000 main-sample fields and 2,000 replication fields (each with ranks, all scores and representation counts), the subset lists, every Section 4.6 field with $r_5\ge2$ or $r_7\ge2$ with its scores, and the four Section 4.5 groups with their point counts. forms.py is a dependency-free exact checker for $\ell$-rank certificates.

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