Properties

 Label 287490q1 Conductor $287490$ Discriminant $-5.694\times 10^{29}$ j-invariant $$-\frac{69557303652860874433300957}{11241080443431191524147200}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -317011292, -36369933689904]) # or

sage: E = EllipticCurve("287490q1")

gp: E = ellinit([1, 1, 0, -317011292, -36369933689904]) \\ or

gp: E = ellinit("287490q1")

magma: E := EllipticCurve([1, 1, 0, -317011292, -36369933689904]); // or

magma: E := EllipticCurve("287490q1");

$$y^2 + x y = x^{3} + x^{2} - 317011292 x - 36369933689904$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{72261192040274987957051582177752}{930316874476975852409673601}, \frac{571803084055612283549149441308307719714471896556}{28375684562306998799034370259333850129599}\right)$$ $$\hat{h}(P)$$ ≈ $68.30502248285609$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(36312, -18156\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(36312, -18156\right)$$

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: $$-569394447701120144272628121600$$ = $$-1 \cdot 2^{26} \cdot 3^{19} \cdot 5^{2} \cdot 7^{8} \cdot 37^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{69557303652860874433300957}{11241080443431191524147200}$$ = $$-1 \cdot 2^{-26} \cdot 3^{-19} \cdot 5^{-2} \cdot 7^{-8} \cdot 13327^{3} \cdot 30859^{3}$$ 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: $$68.3050224829$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0129507403595$$ 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: $$16$$  = $$2\cdot1\cdot2\cdot2\cdot2$$ 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)

Modular invariants

Modular form 287490.2.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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 491692032 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$3.5384024457$$

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{26}$$ Non-split multiplicative 1 1 26 26
$$3$$ $$1$$ $$I_{19}$$ Non-split multiplicative 1 1 19 19
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$2$$ $$I_{8}$$ Non-split multiplicative 1 1 8 8
$$37$$ $$2$$ $$III$$ Additive -1 2 3 0

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 287490q 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
$2$ $$\Q(\sqrt{-111})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$4$ 4.2.243134400.4 $$\Z/4\Z$$ 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.