# Properties

 Label 412224.bs1 Conductor $412224$ Discriminant $1.326\times 10^{24}$ j-invariant $$\frac{13403946614821979039929}{5057590268826067968}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -31672545, -40482638721])

gp: E = ellinit([0, 1, 0, -31672545, -40482638721])

magma: E := EllipticCurve([0, 1, 0, -31672545, -40482638721]);

$$y^2=x^3+x^2-31672545x-40482638721$$

## Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-4182, 137295\right)$$ $\hat{h}(P)$ ≈ $1.7717415635108328010931944392$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-1357, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-4182,\pm 137295)$$, $$\left(-1357, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$412224$$ = $2^{6} \cdot 3 \cdot 19 \cdot 113$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $1325816943431140761403392$ = $2^{28} \cdot 3^{13} \cdot 19 \cdot 113^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{13403946614821979039929}{5057590268826067968}$$ = $2^{-10} \cdot 3^{-13} \cdot 7^{3} \cdot 19^{-1} \cdot 113^{-4} \cdot 3393487^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.3282286353968726977752591608\dots$ Stable Faltings height: $2.2885078645569547336494109786\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.7717415635108328010931944392\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.065683876606557602335084637222\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $208$  = $2^{2}\cdot13\cdot1\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ 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[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $6.0514924202904590163827093390231798924$

## Modular invariants

Modular form 412224.2.a.bs

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{3} - 4q^{5} - 4q^{7} + q^{9} - 4q^{15} - 6q^{17} - q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 121405440 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{18}^{*}$ Additive 1 6 28 10
$3$ $13$ $I_{13}$ Split multiplicative -1 1 13 13
$19$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$113$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## 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
$2$ 2B 8.6.0.4

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 412224.bs consists of 2 curves linked by isogenies of degree 2.