## 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. **Rational points carry the signal: small ones the mean and most of the tail, moderate-height ones 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 instead finds rational points of moderate height on $E_{-16D}$ (denominator $\le5$) 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. The residual tail is thus mostly the same point-detection effect at larger heights.
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% ($p=0.024$) and ×3.0 in the top 0.1% ($p=0.017$), about a third of the $\ell=3$ strength. This sample was a confirmatory test of a hypothesis raised by the main sample. Including the main sample gives ×1.5 ($p=0.008$) and ×3.3 ($p=0.003$), but that pooling mixes hypothesis generation with confirmation. Its source is 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 The residual tail is mostly rational points of moderate height

Section 4.3 leaves open where the residual tail comes from. A rational point of $E_{-16D}$ with $x=X/c^2$ is the same as a representation $4m^3=y^2+Dc^6$, i.e. $z=c^3$, because $E_{-16Dc^6}\cong E_{-16D}$. We searched for such points with $c=1$ ($m\le10^6$), $c=2$ ($m\le4\cdot10^6$), $c=3$ ($9\cdot10^6$), $c=4$ ($1.6\cdot10^7$) and $c=5$ ($2.5\cdot10^7$) in four groups of main-sample fields, all without a $z=1$ point with $m\le3\cdot10^4$. 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 rational point of $E_{-16D}$ with denominator $c\le5$ in the stated range. 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. The residual tail of Section 4.3 is therefore not a separate mechanism. It is the same point-detection effect, at heights beyond the bounded small representations of Table 4.

### 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: ×1.49 in the top 1% ($p=0.024$) and ×3.0 in the top 0.1% ($p=0.017$), against ×4.6 and ×13.0 for $\ell=3$ in the same sample. Including the hypothesis-generating main sample gives ×1.5 (37 against 24.0 expected, $p=0.008$) and ×3.3 (8 against 2.4, $p=0.003$); because the main sample suggested the effect, this pooled figure overstates the evidence. Fields with $r_5=1$ and no point show no tail excess (exceedance ratio 1.03). We also searched the 30 top-1% residual fields with $r_5\ge2$ and 200 low-scoring fields with $r_5=2$ for rational points of moderate height on $y^2=x^5-4^4D$ (denominator $\le3$, $m\le6\cdot10^6$) and found none in either group. For a genus-2 curve, rank comes mostly from degree-2 divisors rather than points on the curve, which this search does not reach, so the source of the $\ell=5$ residual is not identified. For $\ell=7$ no residual is detected (493 fields with $r_7=2$).

## 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 rational points with denominator $\le5$ in 80% of the fields with $r_3=2$, proving positive rank; low-scoring fields with $r_3=2$ have them in only 5%. The association between the score and $r_3$ is therefore mediated by 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, at $p\approx10^{-2}$–$10^{-3}$.
* Representation counts are bounded ($m\le M$, $z\le13$), and Mordell–Weil ranks were bounded below only by points found with denominator $\le5$. 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, and `hp_test.py l5` prints the $\ell=5$ point test (about 1.5 hours). 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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