# Properties

 Label 441090.bw4 Conductor $441090$ Discriminant $-4.098\times 10^{22}$ j-invariant $$\frac{4223169036960119}{11647532812500}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, 5121936, -8659529852])

gp: E = ellinit([1, -1, 0, 5121936, -8659529852])

magma: E := EllipticCurve([1, -1, 0, 5121936, -8659529852]);

## Simplified equation

 $$y^2+xy=x^3-x^2+5121936x-8659529852$$ y^2+xy=x^3-x^2+5121936x-8659529852 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z+5121936xz^2-8659529852z^3$$ y^2z+xyz=x^3-x^2z+5121936xz^2-8659529852z^3 (dehomogenize, simplify) $$y^2=x^3+81950973x-554127959554$$ y^2=x^3+81950973x-554127959554 (homogenize, minimize)

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(3312, 209594\right)$$ (3312, 209594) $\hat{h}(P)$ ≈ $2.3740296425136742031652269552$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{5123}{4}, -\frac{5123}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(3312, 209594\right)$$, $$\left(3312, -212906\right)$$, $$\left(6224, 510986\right)$$, $$\left(6224, -517210\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$441090$$ = $2 \cdot 3^{2} \cdot 5 \cdot 13^{2} \cdot 29$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-40984683415027157812500$ = $-1 \cdot 2^{2} \cdot 3^{8} \cdot 5^{8} \cdot 13^{10} \cdot 29$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4223169036960119}{11647532812500}$$ = $2^{-2} \cdot 3^{-2} \cdot 5^{-8} \cdot 13^{-4} \cdot 29^{-1} \cdot 161639^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.0225211849495534400515450539\dots$ Stable Faltings height: $1.1907403618847302263271787147\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.3740296425136742031652269552\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.058910178299575093192507602174\dots$ 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: $256$  = $2\cdot2^{2}\cdot2^{3}\cdot2^{2}\cdot1$ 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: $L'(E,1)$ ≈ $8.9506886098532522329779929916$

## Modular invariants

Modular form 441090.2.a.bw

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);

$$q - q^{2} + q^{4} + q^{5} + 4 q^{7} - q^{8} - q^{10} - 4 q^{11} - 4 q^{14} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 66060288 $\Gamma_0(N)$-optimal: no Manin constant: 1 (conditional*)
* The Manin constant is correct provided that curve 441090.bw3 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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$3$ $4$ $I_{2}^{*}$ Additive -1 2 8 2
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$13$ $4$ $I_{4}^{*}$ Additive 1 2 10 4
$29$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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 4.6.0.1

## $p$-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 and 4.
Its isogeny class 441090.bw consists of 4 curves linked by isogenies of degrees dividing 4.