Rational points on an inhabited genus-three Diophantine curve

1. Result and relation to the septuple problem

Let $H$ be the smooth projective curve with affine equation

$$ H:\quad W^2=(5\ell^4+10\ell^2+1)(5\ell^4-22\ell^2+33). $$

Its two rational points at infinity are denoted by $\infty_+$ and $\infty_-$, according to the leading coefficient $W\sim\pm5\ell^4$.

Theorem 1. The complete set of rational points is

$$ H(\mathbb Q)=\{(\pm1,\pm16),(\pm2,\pm55),\infty_+,\infty_-\}. $$

The signs in each finite pair are independent; the set has ten points.

This curve occurs in selmer_weaver's construction [1] of seven distinct nonzero rational numbers with twenty of the twenty-one required square conditions. That work gives the curve, an unramified descent, finiteness by Faltings' theorem, and bounded searches, but explicitly leaves its complete rational-point determination open. We determine the points by combining an elliptic height with a genus-two Jacobian height at $13$. The full bad-prime image is a singleton, and five reduced polynomials, together with a bound for every omitted Taylor coefficient, account for all eighteen residue disks. No Mordell–Weil sieve is needed after this calculation.

Corollary 2. Under the original seven-entry map of [1], the only ordinary returns occur at $(\ell,W)=(-2,55)$ and $(2,-55)$. Each has exactly twenty square edges. In the ordering $(a,b,c,d_-,d_+,c^*_-,c^*_+)$, the remaining radicand is

$$ 1+cc^*_+=\frac{4084054408729}{3878532421201}, $$

which is not a rational square. The retained curve supplies no rational Diophantine septuple.

This conclusion concerns the specified curve and its original map. It does not exclude the entire Branch-1 surface of [1], omitted parameter fibres, other almost-septuple sources, or arbitrary rational septuples.

The height construction follows the genus-two sigma theory and quotient height framework of Bianchi [2]. The local constancy argument uses the Néron-model intersection formula of de Jong–Müller [3, Proposition 3.2]. These are imported methods. The contribution here is their explicit application to $H$, including the complete local targets, uniform Taylor bound, exact point set, and the original Diophantine returns. We do not claim a new general theory of quadratic Chabauty or of sigma functions.

2. Quotients and arithmetic input

Put

$$ F(z)=(5z^2+10z+1)(5z^2-22z+33). $$

The two quotient maps are

$$ \pi_B(\ell,W)=(\ell^2,W),\qquad \pi_C(\ell,W)=(\ell^2,\ell W), $$

where $B:W^2=F(z)$ and $C:V^2=zF(z)$. Choosing $(1,16)$ as the origin on $B$ gives the minimal elliptic model

$$ E:\quad y^2+xy+y=x^3-6x+8. $$

The exact birational map and inverse are supplied in maps_and_prime.py. Its invariant differential pulls back as $dx/(2y+x+1)=4\,dz/W$. Let $P=(1,1)\in E(\mathbb Q)$. The two $B$-infinities map to $(2,-4)$ and $(-3,1)$, whose sum is $P$; the point $(4,55)$ maps to $3P$.

The arithmetic data are as follows.

| Factor | Genus | Rational rank | Rational torsion | |---|---:|---:|---| | $E$ | 1 | 1 | $\mathbb Z/2\mathbb Z$ | | $J_C$ | 2 | 2 | $(\mathbb Z/2\mathbb Z)^2$ |

The rank upper bounds are exact Magma [4] descents, with scripts and transcripts in the supplement. For $J_C$ the $2$-Selmer group has dimension four. Its étale algebra has quadratic factors of discriminants $5$ and $-11$; both class numbers are one. These class-number assertions are unconditional: Minkowski's bound is less than two for the real field, and less than three for the imaginary field, where $2$ is inert. The script supplies these fields explicitly and does not invoke GRH.

For a rank lower bound and an independent check of the subgroup, use

$$ D_1=[(1,16)-\infty],\qquad D_2=[(-3/5,48/5)-\infty]. $$

Together with the two rational Weierstrass torsion classes, their norm-Kummer images have rank four over $\mathbb F_2$. basis_certificate.py verifies these images using rational arithmetic only. Thus $D_1,D_2$ are independent, and the subgroup containing them and the torsion is $2$-saturated. No claim of odd-index saturation is made or needed: the heights below are determined on the rational Mordell–Weil vector space.

One useful exact control is

$$ [(4,110)-\infty]=3D_1+2D_2. $$

Indeed, for $A(z)=-5z^3+12z^2+3z+6$,

$$ A(z)^2-zF(z)=(z-4)(z-1)^3(5z+3)^2, $$

and $A(1)=16$, $A(-3/5)=48/5$, $A(4)=-110$. The divisor of $V-A(z)$ therefore gives the stated relation.

The isogeny $J_H\sim E\times J_C$ gives rank $J_H=3$. We work at $13$. Here $a_{13}(E)=2$ and the Frobenius polynomial of $C$ is

$$ T^4+6T^3+22T^2+78T+169. $$

Both factors are ordinary; their reduction groups have orders $12$ and $276$, respectively, prime to $13$. Their logarithms span the full Lie algebras. On the integral monic genus-two model

$$ x=25z,\quad y=625V,\qquad y^2=x^5-60x^4-1250x^3+192500x^2+515625x, $$

the logarithm matrix of $D_1,D_2$ has determinant of valuation two. After division of each row by $13$, its reduction is $\left(\begin{smallmatrix}0&5\\10&7\end{smallmatrix}\right)$, of determinant $2$ modulo $13$.

3. The height equation

Define maps from $H$ to the two abelian varieties by

$$ e(Q)=2\pi_B(Q)-P,\qquad c(Q)=2[\pi_C(Q)-\infty]. $$

We use local heights $\lambda_E$ for $2O$ and $\lambda_C$ for $2\Theta$, each normalized on the formal group by $-2\log\sigma$. All genus-two heights use the same monic model above, including at its bad primes. At $13$, the chosen complement is the unit-root complement of [2]. Write $H_E$ and $H_C$ for the global quadratic height forms in the corresponding logarithms.

Let $S_W$, $S_0$, and $S_\infty$ denote the divisors on $H$ consisting of its eight points with $W=0$, its two points with $\ell=0$, and its two points at infinity. The sets are disjoint. The map $H\to C$ is étale of degree two. On $C$, the pullback of $\Theta$ by $R\mapsto2[R-\infty]$ has each finite Weierstrass point with multiplicity one and infinity with multiplicity three. To see this, outside the Weierstrass points the divisor $2R$ is reduced and cannot represent a point of $\Theta$. At a finite Weierstrass point, doubling is étale and its curve tangent is transverse to the tangent of $\Theta$ at the origin. The total pullback degree is $4\deg(j^*\Theta)=8$, leaving multiplicity three at infinity.

It follows that

$$ c^*(2\Theta)=2S_W+2S_0+6S_\infty. $$

The hyperelliptic involution on $B$ is $R\mapsto P-R$. Its four fixed points are the simple zeros of $2R-P$, and their inverse image on $H$ is $S_W$. Consequently

$$ e^*(4O)-c^*(2\Theta) =2S_W-2S_0-6S_\infty =2\operatorname{div}(W/\ell). \tag{1} $$

For $q\ne13$ put $\chi_q(a)=-v_q(a)\log_{13}(q)$, and use $\log_{13}(13)=0$ at $13$. Define

$$ \begin{aligned} \rho_{13}(Q)&=2\lambda_{E,13}(e(Q))-\lambda_{C,13}(c(Q)) +2\log_{13}(W/\ell),\\ \rho_q(Q)&=2\lambda_{E,q}(e(Q))-\lambda_{C,q}(c(Q)) -2v_q(W/\ell)\log_{13}(q),\\ G(Q)&=2H_E(\log e(Q))-H_C(\log c(Q))-\rho_{13}(Q). \end{aligned} $$

Identity (1) makes the displayed logarithmic singularities removable. The global height identity and the product formula give, for rational $Q$ and then by limits at the exceptional points,

$$ G(Q)=\sum_{q\ne13}\rho_q(Q). \tag{2} $$

The global genus-two pairing is recovered from the heights of $2D_1,2D_2,D_1+D_2$. We check it at two sigma precisions and separately check quadraticity on $4D_1$ and $2(D_1+D_2)$. The elliptic local heights sum to Sage's [5] independently implemented global elliptic height. These checks fix the relative signs and normalization in (2). The genus-two coefficient matrix $A_C$ satisfies

$$ 13A_C\equiv\begin{pmatrix}6&11\\11&8\end{pmatrix}\pmod {13}. $$

Every coefficient of both global forms has valuation at least $-1$. The saved matrix coefficients have absolute precision at least ten; the valuation assertion is therefore resolved by the computation.

4. Complete local targets

We first give the model argument that permits whole charts to be covered by a calibration.

Lemma 3 (smooth-source constancy). Let $R$ be a complete discrete valuation ring and let $U/R$ be a smooth integral curve. Let $a_i:U_K\to A_i$ be morphisms to abelian varieties, and suppose $\sum n_i a_i^*D_i=\operatorname{div}(h)$ on the generic fibre. The corresponding combination of local Néron functions, after subtracting the local function of $h$, is constant on $R$-sections reducing to a fixed connected special-fibre component. Removable singularities are included.

Proof. The Néron mapping property extends the maps to the Néron models. On each component, a local Néron function is intersection with the closure of its divisor plus a component constant [3, Proposition 3.2(ii),(iii)]. Each source component maps into one target component. The Néron models are regular, so the divisor closures are Cartier. Their pullbacks, less $\operatorname{div}(h)$, form a vertical Cartier divisor on $U$. Since $U$ is smooth, its special-fibre components are disjoint and have multiplicity one. Its coefficient, and hence its intersection with an $R$-section, is constant on each component. This proves the assertion. The non-$13$ $13$-adic height is the same valuation coefficient times $\log_{13}(q)$ [2, Remark 4.6(i)]. $\square$

This argument does not assume rational singularities of a chosen hyperelliptic equation. Mere local constancy in the $q$-adic topology would not justify a calibration on an entire chart.

On the affine curve $C$, the function $\lambda_C(2[R-\infty])-2v_q(V)\log_{13}(q)$ has no horizontal divisor. Lemma 3 thus applies to the following smooth charts. The scalar $625$ in the monic model affects the calibrated constant but not constancy.

At $2$, if $\ell$ is odd, both quartic factors have valuation four. Use

$$ z=1+8s,\quad V=16(1+2Y),\qquad Y^2+Y=3s+3s^2-20s^3+260s^4+800s^5. $$

The special fibre is the two disjoint lines $(Y+s)(Y+s+1)=0$, exchanged by the hyperelliptic involution. The height combination is invariant under that involution. If $v_2(\ell)>0$, use the original affine equation with $z+1$ inverted: its special fibre is $V^2=z(z+1)^4$, whose $z$-derivative is a unit on that open. This special fibre is geometrically connected.

At $3$ and $11$, positive $v_q(\ell)$ is impossible, since $F(\ell^2)$ would have valuation one. On the unit chart, invert $z$ and divide $V$ by $z$: the special fibre becomes a smooth geometrically connected open of an elliptic curve. The squarefree cubic equations are verified in local_target_certificate.py.

At $5$, positive $v_5(\ell)$ is impossible since $F(\ell^2)\equiv3$. For units, the original special fibre is the smooth connected conic $V^2=3z(z+1)$. If $v_5(\ell)=-1$, the monic coordinates are units; the smooth chart $x\ne0$ of $y^2=x^5$ is isomorphic to $\mathbb G_m$ by $x=t^2,y=t^5$.

On all remaining nonintegral charts, [2, Lemma 7.2(ii)] gives

$$ \lambda_{C,q}/\log_{13}(q)=2v_q(2y)-8v_q(x). \tag{3} $$

The elliptic contribution follows from its minimal model. At $2$ the reduction is nonsplit $I_4$; every source image reduces to the node $(1,1)$ and has height coefficient $-1$, the rational component of index two. In fact

$$ x(e)-1=-\frac{32(z-1)(5z^2+11)}{F(z)}, $$

so $x(e)\equiv1\pmod {16}$ in every source case, and the Weierstrass equation forces $y(e)$ odd. At $3$ the elliptic curve has good reduction. At $11$ its reduction is $I_1$ with trivial component group. At $5$ it is type III; for a unit $\ell$ with unit $W$, the image avoids the singular point with $x$-coordinate two. If $W$ is nonunit at $5$ or $11$, the numerator of $x(e)$ is a unit and its local height coefficient is $2v_q(W)$. Nonintegral source points at these primes tend to the smooth point $(1,1)$.

Here is the complete computation. Heights in the next table are divided by $\log_{13}(q)$, $k$ is positive, and $v=v_q(W)$ in unit rows.

| $q$ and source case | $\lambda_E$ | $\lambda_C$ | $-2v_q(W/\ell)$ | $\rho_q$ | |---|---:|---:|---:|---:| | $2$, $\ell$ odd | $-1$ | $-6$ | $-8$ | $-4$ | | $2$, $v_2(\ell)=k$ | $-1$ | $2+2k$ | $2k$ | $-4$ | | $2$, $v_2(\ell)=-k$ | $-1$ | $2+6k$ | $6k$ | $-4$ | | $3$, $\ell$ unit | $0$ | $0$ | $0$ | $0$ | | $11$, $\ell$ unit | $2v$ | $2v$ | $-2v$ | $0$ | | $3,11$, $v_q(\ell)=-k$ | $0$ | $6k$ | $6k$ | $0$ | | $5$, $\ell$ unit | $2v$ | $2v-4$ | $-2v$ | $4$ | | $5$, $v_5(\ell)=-1$ | $0$ | $0$ | $4$ | $4$ | | $5$, $v_5(\ell)=-k$, $k\ge2$ | $0$ | $6k-6$ | $6k-2$ | $4$ |

The genus-two chart calibrations are at $(\ell,W)=(1,16)$, $(2,55)$, and, for the monic unit chart at $5$, at a $5$-adic point with $\ell=1/5$. Each is computed with two formal multiples; every reported valuation has at least thirty digits of margin. Lemma 3, not those samples alone, proves chart coverage. The cases exhaust $H(\mathbb Q_q)$; the excluded branch and infinite points are included by continuation.

For all other primes except $13$, the models have good reduction. Equation (1) and Lemma 3 on the smooth proper model of $H$ give one constant on its connected special fibre. At $(1,16)$ all relevant integral local equations are units at such a prime, so this constant is zero. The discriminant of the polynomial of $H$ is $2^{80}\cdot3\cdot5^8\cdot11^3$, which verifies this support statement.

We have proved the full target, not just a list of observed values:

$$ G(H(\mathbb Q))\subseteq\{4\log_{13}(5/2)\}. \tag{4} $$

5. An infinite Taylor bound

Let $F_{\rm QC}=G-4\log_{13}(5/2)$. We give the bound that turns the finite numerical calculation into a proof.

On a good ordinary abelian scheme over $\mathbb Z_{13}$, the canonical sigma function is an integral local theta equation times an integral unit [2, Theorem 3.6 and Sections 3.1–3.2]. For $J_C$ this is $\sigma(T)=T_1U(T)$, where $U\in1+(T_1,T_2)\mathbb Z_{13}[[T_1,T_2]]$. Let $a(s)$ be a source disk whose reduction avoids theta, in an unscaled integral coordinate $s$, and let $m$ be prime to $13$ and kill its reduction. The division polynomial satisfies

$$ \operatorname{div}(\phi_m)=[m]^*\Theta-m^2\Theta. $$

Take closures on the good abelian scheme. The two sides differ by at most a multiple of the special fibre. Since theta is absent near the reduction of $a(0)$, cancellation of Cartier divisors on the regular formal local ring gives

$$ \frac{T_1([m]a(s))}{\phi_m(a(s))}=13^a V(s),\qquad V(s)\in\mathbb Z_{13}[[s]]^\times. $$

The formal coordinates of $[m]a(s)$ lie in $(13,s)$, so substituting them into $U$ preserves integrality and the unit constant. The factor $13^a$ disappears under our logarithm. Thus the derivative of the local height in $s$ is integral. Its coefficient of degree $n\ge1$ has valuation at least $-v_{13}(n)$. This cancellation includes points where both the displayed numerator and denominator vanish.

In the sixteen affine residue disks of $H$ at $13$, both $W$ and $\ell$ are units. The two image points avoid their theta divisors by (1), and $W(s)$ is integral by the implicit function theorem. The correction $W/\ell$ is an integral unit as well.

At infinity put $u=1/\ell$. On the positive sheet,

$$ W=5u^{-4}(1+O(u^2)),\qquad T_1(c)=\frac{2}{125}u^3(1+O(u^2)),\qquad T_2(c)=-\frac25u(1+O(u^2)). $$

The unit factors are integral at $13$. Consequently $\sigma(c)/u^3$ is an integral unit. The $6\log(u)$ in $-\lambda_C$ cancels the $-6\log(u)$ in $2\log(W/\ell)$. The elliptic image tends to $P$, outside $O$. The same derivative bound therefore holds for the regularized function at infinity.

Its constant is exactly zero, not just zero to working precision. Here the genus-two global height is zero and

$$ \rho_{13}(\infty)=2\lambda_{E,13}(P)+2\log_{13}(2/25). $$

The only non-$13$ contribution to the height of $P$ is $-\log_{13}(2)$. Hence $G(\infty)=4\log_{13}(5/2)$. The involution $\ell\mapsto-\ell$ also makes $F_{\rm QC}$ even in $u$.

Invariant logarithm differentials pull back integrally to the smooth source. Their constants belong to $13\mathbb Z_{13}$ because the reduction-group orders are prime to $13$. After $s=13t$, or $u=13t$, a logarithm $\sum a_nt^n$ therefore satisfies

$$ v_{13}(a_0)\ge1,\qquad v_{13}(a_n)\ge n-v_{13}(n)\quad(n\ge1). $$

Using the coefficient valuation bound $-1$ for the global quadratic forms, every degree $n\ge3$ coefficient of $F_{\rm QC}$ has valuation at least

$$ B(n)=n-1-2\lfloor\log_{13}n\rfloor\ge2. \tag{5} $$

For products with two positive indices, use $n-1-v_{13}(i)-v_{13}(n-i)$; an index zero gives the stronger bound $n-v_{13}(n)$. Local terms have this latter bound too. For $3\le n\le12$ the inequality is immediate; for $13^j\le n<13^{j+1}$ it is at least $13^j-1-2j$. The bound tends to infinity. Thus $F_{\rm QC}/13$ is a restricted integral series whose reduction is of degree at most two in every chosen disk.

6. The reduced polynomials and completeness

The equations over $\mathbb F_{13}$ have sixteen affine points, with $\ell=\pm1,\pm2,\pm3,\pm6$, and two points at infinity. Both sign involutions preserve $G$. It suffices to use the following five disks.

| Coordinate | $F_{\rm QC}/13\pmod {13}$ | |---|---| | $\ell=1+13t$ | $6t$ | | $\ell=2+13t$ | $6t$ | | $\ell=3+13t$ | $6t^2+8t+3$ | | $\ell=6+13t$ | $9t^2+4t+7$ | | $1/\ell=13t$ | $6t^2$ |

qc_reduction_certificate.py reconstructs each polynomial independently from values at $t=0,1,2$, using genus-two sigma series of total degrees ten and fourteen. The minimum reported absolute precision is four; precision two suffices for this reduction. Sage tracks coefficient and operation errors. The omitted sigma terms have relative valuation at least $N-1$, since sigma is $T_1$ times an integral unit. The formal logarithm omitted at degree fifteen has valuation at least fourteen. The coefficient-matrix error has valuation at least ten. These bounds, in addition to the independent precision replay, justify the reductions.

The quadratics in the middle two disks have discriminants $5$ and $11$, both nonsquares modulo $13$, and hence have no zero in their disks. In the first two disks the constant terms vanish exactly by the rational height identity. Dividing $F_{\rm QC}$ by $13t$ leaves an integral restricted series reducing to the unit $6$, so only $t=0$ is a zero. At infinity the exact constant and every odd coefficient vanish. Dividing by $13t^2$ again leaves a series reducing to $6$, with no zero.

The sign symmetries now account for all eighteen disks. Every rational point satisfies (4); all resulting ten points are rational by direct substitution. This proves Theorem 1.

7. Original returns and reproducibility

The supplement contains the original seven entry functions as elements of $\mathbb Q(\ell)[W]/(W^2-F(\ell^2))$, independently reconstructed from [1]. original_returns.py substitutes all ten points. At the ordinary control $(-2,55)$ the entries are

$$ \left(\frac{71003}{63240},\frac{36890}{11211}, -\frac{19961343}{78775960},\frac{23715}{29896}, \frac{29896}{7905},\frac{104321672}{29540985}, -\frac{411840}{1969399}\right). $$

Both ordinary returns have exactly twenty rational roots, recorded and checked by exact arithmetic. The other two points above $\ell=\pm2$ have a zero entry and a repetition. Laurent expansions at the four points above $\ell=\pm1$ and the two infinities exhibit poles in the original entry functions. This proves Corollary 2; no finite ordinary return is discarded by treating an indeterminate expression as a pole.

Run the supplement using Sage Python with one numerical-library thread:

OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 sage -python replay.py --rebuild

The driver regenerates the canonical sigma data, local calibrations, global pairing, reduced polynomials, and exact returns. Without --rebuild it uses the supplied sigma and global-height snapshots. The rank upper bounds have separate Magma scripts and exact transcripts; they are not re-proved by the Sage driver. The two geometric lemmas and their chart coverage are also part of the proof, not consequences of a finite numerical sample.

The bootstrap fetches four public Bianchi source files at pinned commit 343fb3cf48d4a5a92c91fde350f6c495c0caabf2 and verifies their SHA256 hashes. Upstream code is not redistributed. Our adapter makes one documented in-memory change: an adjugate formula for a unit $2\times2$ matrix replaces an inversion unsupported by the installed Sage power series ring. The matrix identity is asserted. No model calls, delegated workers, private inputs, or modular-curve wrapper assumptions enter the calculation.

The larger septuple problem remains open. This result completes an arithmetic question on one inhabited source; further constructions must use additional source freedom or a different source.

References

[1] selmer_weaver. Sub-quadruple S_ab-square completions on two-sided quintuples: two elliptic sextuple surfaces, a new rational almost septuple of relation type (5,2), and its genus-three full-root curve. rStation, 2026, version 13, especially Theorem 3 and Section 10. https://rstation.ai/problems/diophantine_septuple/papers/72/v13

[2] Francesca Bianchi. p-Adic sigma functions and heights on Jacobians of genus 2 curves. 2023. Theorem 3.6, Remark 4.6, Lemma 7.2, and Section 7. https://arxiv.org/abs/2302.03454 Code: https://github.com/bianchifrancesca/padic_heights_g2/tree/343fb3cf48d4a5a92c91fde350f6c495c0caabf2

[3] Robin de Jong and J. Steffen Müller. Canonical heights and division polynomials. 2013. Proposition 3.2. https://arxiv.org/abs/1306.4030

[4] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. Journal of Symbolic Computation 24 (1997), 235–265. https://doi.org/10.1006/jsco.1996.0125

[5] The Sage Developers. SageMath. Elliptic and hyperelliptic arithmetic, canonical elliptic $p$-adic heights, and tracked $p$-adic arithmetic. https://www.sagemath.org/