Show commands: Magma / Pari/GP / SageMath

Frey curve for $3 + 125 = 128$ (factored form: $3 + 5^3 = 2^7$)

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -334, -2368])

gp: E = ellinit([1, 0, 1, -334, -2368])

magma: E := EllipticCurve([1, 0, 1, -334, -2368]);

## Simplified equation

 $$y^2+xy+y=x^3-334x-2368$$ y^2+xy+y=x^3-334x-2368 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-334xz^2-2368z^3$$ y^2z+xyz+yz^2=x^3-334xz^2-2368z^3 (dehomogenize, simplify) $$y^2=x^3-432243x-109173042$$ y^2=x^3-432243x-109173042 (homogenize, minimize)

## Mordell-Weil group structure

$$\Z/{2}\Z \oplus \Z/{2}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-11, 5\right)$$, $$\left(21, -11\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-11, 5\right)$$, $$\left(21, -11\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$30$$ = $2 \cdot 3 \cdot 5$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $9000000$ = $2^{6} \cdot 3^{2} \cdot 5^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4102915888729}{9000000}$$ = $2^{-6} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{3} \cdot 2287^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.21759239493637902558062626083\dots$ Stable Faltings height: $0.21759239493637902558062626083\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $1.1173160864138321498160760446\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $8$  = $2\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar/factorial(ar)  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $0.55865804320691607490803802232$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 12 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$5$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6

## Galois representations

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.12.0.1
$3$ 3B.1.2 3.8.0.2

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 5 nonsplit split nonsplit 0 1 0 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 30a consists of 8 curves linked by isogenies of degrees dividing 12.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/2\Z \oplus \Z/6\Z$$ 2.0.3.1-300.1-a6 $3$ 3.1.243.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{-5})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{5})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $6$ 6.0.177147.2 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.207360000.1 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $8$ 8.0.12960000.1 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $12$ 12.0.8226356490141696.17 $$\Z/6\Z \oplus \Z/12\Z$$ Not in database $16$ 16.0.11007531417600000000.1 $$\Z/4\Z \oplus \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/24\Z$$ Not in database $18$ 18.0.617673396283947000000000000.3 $$\Z/2\Z \oplus \Z/18\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

This is a Frey curve corresponding to the ABC triple $(3,125,128) = (3,5^3,2^7)$ associated to the largest solution of the $S$-unit equation for $S = \{2,3,5\}$. Because the valuation at $2$ of $ABC$ exceeds $4$, there is a quadratic twist with multiplicative reduction at $2$, so the conductor is a multiple of $2$ but not of $4$.