# Properties

 Label 287490cw1 Conductor $287490$ Discriminant $-2.678\times 10^{16}$ j-invariant $$\frac{229012002719}{14288400000}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 15715, 7843187]) # or

sage: E = EllipticCurve("287490cw1")

gp: E = ellinit([1, 1, 1, 15715, 7843187]) \\ or

gp: E = ellinit("287490cw1")

magma: E := EllipticCurve([1, 1, 1, 15715, 7843187]); // or

magma: E := EllipticCurve("287490cw1");

$$y^2 + x y + y = x^{3} + x^{2} + 15715 x + 7843187$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(237, 4876\right)$$ $$\hat{h}(P)$$ ≈ $0.1089603012666046$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-133, 1916\right)$$, $$\left(-133, -1784\right)$$, $$\left(-33, 2716\right)$$, $$\left(-33, -2684\right)$$, $$\left(77, 3046\right)$$, $$\left(77, -3124\right)$$, $$\left(237, 4876\right)$$, $$\left(237, -5114\right)$$, $$\left(607, 15236\right)$$, $$\left(607, -15844\right)$$, $$\left(2235, 104776\right)$$, $$\left(2235, -107012\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$287490$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 37^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-26778762032400000$$ = $$-1 \cdot 2^{7} \cdot 3^{6} \cdot 5^{5} \cdot 7^{2} \cdot 37^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{229012002719}{14288400000}$$ = $$2^{-7} \cdot 3^{-6} \cdot 5^{-5} \cdot 7^{-2} \cdot 19^{3} \cdot 29^{3} \cdot 37^{2}$$ 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: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.108960301267$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.286115801198$$ 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: $$420$$  = $$7\cdot2\cdot5\cdot2\cdot3$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 287490.2.a.cw

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);

$$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + q^{10} + 3q^{11} - q^{12} + 3q^{13} - q^{14} - q^{15} + q^{16} + 4q^{17} + q^{18} + 7q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3144960 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$13.0936108362$$

## 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$$(j)_{-}$$
$$2$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$3$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$5$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$37$$ $$3$$ $$IV$$ Additive 1 2 4 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

## $p$-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 no rational isogenies. Its isogeny class 287490cw consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.54760.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.119946304000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.