Properties

 Label 422730.ey1 Conductor $422730$ Discriminant $4.276\times 10^{21}$ j-invariant $$\frac{13540496380264015726564729}{5864914552058284800}$$ 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, 1, -44690252, 114960047951])

gp: E = ellinit([1, -1, 1, -44690252, 114960047951])

magma: E := EllipticCurve([1, -1, 1, -44690252, 114960047951]);

$$y^2+xy+y=x^3-x^2-44690252x+114960047951$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(3631, 21669\right)$$ $\hat{h}(P)$ ≈ $0.33621492326647138753281039002$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{15195}{4}, -\frac{15199}{8}\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(2289, 155869\right)$$, $$\left(2289, -158159\right)$$, $$\left(3631, 21669\right)$$, $$\left(3631, -25301\right)$$, $$\left(3981, 9349\right)$$, $$\left(3981, -13331\right)$$, $$\left(4729, 94381\right)$$, $$\left(4729, -99111\right)$$, $$\left(12171, 1166029\right)$$, $$\left(12171, -1178201\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$422730$$ = $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 61$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $4275522708450489619200$ = $2^{8} \cdot 3^{7} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} \cdot 61^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{13540496380264015726564729}{5864914552058284800}$$ = $2^{-8} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-2} \cdot 61^{-6} \cdot 229^{3} \cdot 1040821^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.1103677064066720029900708314\dots$ Stable Faltings height: $2.5610615620726171572924482129\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.33621492326647138753281039002\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.13616681646966807590749580218\dots$ 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: $1536$  = $2^{3}\cdot2^{2}\cdot2\cdot2\cdot2\cdot( 2 \cdot 3 )$ 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: $L'(E,1)$ ≈ $17.580025248303031486999621041100355113$

Modular invariants

Modular form 422730.2.a.ey

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);

$$q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} + q^{11} + 6q^{13} + q^{14} + q^{16} - 2q^{17} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 47185920 $\Gamma_0(N)$-optimal: not computed* (one of 2 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 422730.ey2 is optimal.

Local data

This elliptic curve is not semistable. There are 6 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$(j)_{-}$
$2$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$3$ $4$ $I_1^{*}$ Additive -1 2 7 1
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$11$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$61$ $6$ $I_{6}$ Split multiplicative -1 1 6 6

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
$2$ 2B 2.3.0.1

$p$-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.
Its isogeny class 422730.ey consists of 2 curves linked by isogenies of degree 2.