Properties

 Label 287490.z1 Conductor $287490$ Discriminant $-33959756250$ j-invariant $$-\frac{446081167009}{24806250}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -1767, -30681]) # or

sage: E = EllipticCurve("287490z1")

gp: E = ellinit([1, 1, 0, -1767, -30681]) \\ or

gp: E = ellinit("287490z1")

magma: E := EllipticCurve([1, 1, 0, -1767, -30681]); // or

magma: E := EllipticCurve("287490z1");

$$y^2 + x y = x^{3} + x^{2} - 1767 x - 30681$$

Mordell-Weil group structure

$$\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(93, -834\right)$$ $$\hat{h}(P)$$ ≈ $0.7670872919881362$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(53, 141\right)$$, $$\left(53, -194\right)$$, $$\left(93, 741\right)$$, $$\left(93, -834\right)$$

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: $$-33959756250$$ = $$-1 \cdot 2 \cdot 3^{4} \cdot 5^{5} \cdot 7^{2} \cdot 37^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{446081167009}{24806250}$$ = $$-1 \cdot 2^{-1} \cdot 3^{-4} \cdot 5^{-5} \cdot 7^{-2} \cdot 37 \cdot 2293^{3}$$ 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: $$0.767087291988$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.366967702242$$ 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: $$20$$  = $$1\cdot2\cdot5\cdot2\cdot1$$ 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.z

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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 322560 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$5.6299252192$$

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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$5$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$37$$ $$1$$ $$II$$ Additive 1 2 2 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 287490.z 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.