# Properties

 Label 405720.gn2 Conductor $405720$ Discriminant $4.149\times 10^{19}$ j-invariant $$\frac{168591300897604}{472410225}$$ 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, 0, 0, -5115747, -4442809714])

gp: E = ellinit([0, 0, 0, -5115747, -4442809714])

magma: E := EllipticCurve([0, 0, 0, -5115747, -4442809714]);

$$y^2=x^3-5115747x-4442809714$$

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-1358, 0\right)$$, $$\left(2611, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1358, 0\right)$$, $$\left(-1253, 0\right)$$, $$\left(2611, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$405720$$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $41489195539442918400$ = $2^{10} \cdot 3^{12} \cdot 5^{2} \cdot 7^{8} \cdot 23^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{168591300897604}{472410225}$$ = $2^{2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{3} \cdot 23^{-2} \cdot 2677^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.6365916841917022965016345112\dots$ Stable Faltings height: $0.53670781486336970707030875313\dots$

## 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: $0.10040271025701693477128943254\dots$ 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: $128$  = $2\cdot2^{2}\cdot2\cdot2^{2}\cdot2$ 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: $4$ = $2^2$ (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: $L(E,1)$ ≈ $3.2128867282245419126812618412452695751$

## Modular invariants

Modular form 405720.2.a.gn

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);

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

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 11796480 $\Gamma_0(N)$-optimal: no Manin constant: 1

## 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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$3$ $4$ $I_{6}^{*}$ Additive -1 2 12 6
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $4$ $I_{2}^{*}$ Additive -1 2 8 2
$23$ $2$ $I_{2}$ Split multiplicative -1 1 2 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 4.12.0.1

## $p$-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.
Its isogeny class 405720.gn consists of 2 curves linked by isogenies of degrees dividing 4.