Properties

 Label 227430ge1 Conductor $227430$ Discriminant $-4.494\times 10^{14}$ j-invariant $$-\frac{47832147}{353780}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -8190, 1061120]) # or

sage: E = EllipticCurve("227430ge1")

gp: E = ellinit([1, -1, 0, -8190, 1061120]) \\ or

gp: E = ellinit("227430ge1")

magma: E := EllipticCurve([1, -1, 0, -8190, 1061120]); // or

magma: E := EllipticCurve("227430ge1");

$$y^2 + x y = x^{3} - x^{2} - 8190 x + 1061120$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-122, 544\right)$$ $$\left(-14, 1090\right)$$ $$\hat{h}(P)$$ ≈ $3.4330448657591837$ $1.261919797490537$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-128, 64\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-128, 64\right)$$, $$\left(-122, 544\right)$$, $$\left(-122, -422\right)$$, $$\left(-14, 1090\right)$$, $$\left(-14, -1076\right)$$, $$\left(41, 870\right)$$, $$\left(41, -911\right)$$, $$\left(88, 964\right)$$, $$\left(88, -1052\right)$$, $$\left(233, 3313\right)$$, $$\left(233, -3546\right)$$, $$\left(347, 6144\right)$$, $$\left(347, -6491\right)$$, $$\left(6731, 548784\right)$$, $$\left(6731, -555515\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$227430$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-449385078064860$$ = $$-1 \cdot 2^{2} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 19^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{47832147}{353780}$$ = $$-1 \cdot 2^{-2} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-2} \cdot 11^{6} \cdot 19^{-2}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.62900604917$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.45333767579$$ 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: $$32$$  = $$2\cdot2\cdot1\cdot2\cdot2^{2}$$ 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: $$1$$ (rounded)

Modular invariants

Modular form 227430.2.a.g

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 829440 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

Special L-value

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);

$$L^{(2)}(E,1)/2!$$ ≈ $$9.53461993574$$

Local data

This elliptic curve is not semistable.

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
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$19$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 227430ge consists of 2 curves linked by isogenies of degree 2.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$4$ 4.2.3119040.1 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.