# Properties

 Label 444360bm2 Conductor $444360$ Discriminant $7.895\times 10^{21}$ j-invariant $$\frac{394315384276816}{208332909225}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -5131476, -1321603776])

gp: E = ellinit([0, 1, 0, -5131476, -1321603776])

magma: E := EllipticCurve([0, 1, 0, -5131476, -1321603776]);

$$y^2=x^3+x^2-5131476x-1321603776$$

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-261, 0\right)$$, $$\left(2384, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2124, 0\right)$$, $$\left(-261, 0\right)$$, $$\left(2384, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$444360$$ = $2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $7895231340820269062400$ = $2^{8} \cdot 3^{8} \cdot 5^{2} \cdot 7^{4} \cdot 23^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{394315384276816}{208332909225}$$ = $2^{4} \cdot 3^{-8} \cdot 5^{-2} \cdot 7^{-4} \cdot 23^{-2} \cdot 29101^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.8932552495578850128134790831\dots$ Stable Faltings height: $0.86341002122001329446528125289\dots$

## 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[1]  magma: RealPeriod(E); Real period: $0.10645153352609903442166667002\dots$ 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: $512$  = $2^{2}\cdot2^{3}\cdot2\cdot2\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  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[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $3.4064490728351691014933334406459739082$

## Modular invariants

Modular form 444360.2.a.bm

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);

$$q + q^{3} - q^{5} - q^{7} + q^{9} + 4 q^{11} - 2 q^{13} - q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 30277632 $\Gamma_0(N)$-optimal: not computed* (one of 4 curves in this isogeny class which might be optimal) Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that curve 444360bm1 is optimal.

## Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $4$ $I_{1}^{*}$ Additive -1 3 8 0
$3$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$23$ $4$ $I_{2}^{*}$ Additive -1 2 8 2

## 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 8.24.0.14

## $p$-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 $0$.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 444360bm consists of 3 curves linked by isogenies of degrees dividing 8.