# Properties

 Label 411840.i1 Conductor $411840$ Discriminant $1.641\times 10^{18}$ j-invariant $$\frac{912446049969377120252018}{17177299425}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-923790828x-10807085379152$$ y^2=x^3-923790828x-10807085379152 (homogenize, simplify) $$y^2z=x^3-923790828xz^2-10807085379152z^3$$ y^2z=x^3-923790828xz^2-10807085379152z^3 (dehomogenize, simplify) $$y^2=x^3-923790828x-10807085379152$$ y^2=x^3-923790828x-10807085379152 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -923790828, -10807085379152])

gp: E = ellinit([0, 0, 0, -923790828, -10807085379152])

magma: E := EllipticCurve([0, 0, 0, -923790828, -10807085379152]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-17548, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-17548, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$411840$$ = $2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $1641316519880294400$ = $2^{17} \cdot 3^{13} \cdot 5^{2} \cdot 11 \cdot 13^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{912446049969377120252018}{17177299425}$$ = $2 \cdot 3^{-7} \cdot 5^{-2} \cdot 11^{-1} \cdot 13^{-4} \cdot 47^{3} \cdot 1637927^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.4832258685258237328760982412\dots$ Stable Faltings height: $1.9519612183985130321707301173\dots$

## 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: $0.027384583700529909216817240956\dots$ 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: $64$  = $2^{2}\cdot2^{2}\cdot2\cdot1\cdot2$ 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: $L(E,1)$ ≈ $0.43815333920847854746907585529$

## Modular invariants

Modular form 411840.2.a.i

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 88080384 $\Gamma_0(N)$-optimal: no Manin constant: 1 (conditional*)
* The Manin constant is correct provided that curve 411840.i4 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$(j)_{-}$
$2$ $4$ $I_{7}^{*}$ Additive 1 6 17 0
$3$ $4$ $I_{7}^{*}$ Additive -1 2 13 7
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $2$ $I_{4}$ Non-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 8.12.0.7

## $p$-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 411840.i consists of 4 curves linked by isogenies of degrees dividing 4.