Properties

 Label 450840.q2 Conductor $450840$ Discriminant $9.549\times 10^{21}$ j-invariant $$\frac{896581610757188944}{1545359765625}$$ 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, -36863780, -86007863100])

gp: E = ellinit([0, -1, 0, -36863780, -86007863100])

magma: E := EllipticCurve([0, -1, 0, -36863780, -86007863100]);

$$y^2=x^3-x^2-36863780x-86007863100$$

Mordell-Weil group structure

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

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-3615, 0\right)$$, $$\left(7010, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3615, 0\right)$$, $$\left(-3394, 0\right)$$, $$\left(7010, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$450840$$ = $2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $9549114360984900000000$ = $2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 13^{2} \cdot 17^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{896581610757188944}{1545359765625}$$ = $2^{4} \cdot 3^{-4} \cdot 5^{-8} \cdot 7^{3} \cdot 13^{-2} \cdot 17^{-2} \cdot 54667^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.1116976636497256252733450215\dots$ Stable Faltings height: $1.2329928712483207122037562983\dots$

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: $0.061276584442858106569159001735\dots$ 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: $512$  = $2^{2}\cdot2\cdot2^{3}\cdot2\cdot2^{2}$ 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: $L(E,1)$ ≈ $1.9608507021714594102130880555117519446$

Modular invariants

Modular form 450840.2.a.q

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 56623104 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $4$ $I_1^{*}$ Additive -1 3 8 0
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$13$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$17$ $4$ $I_2^{*}$ 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 $\GL(2,\Z_\ell)$ 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 450840.q consists of 2 curves linked by isogenies of degrees dividing 4.