# Properties

 Label 286650b1 Conductor 286650 Discriminant 7546061250 j-invariant $$\frac{590625}{338}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -492, 566]); // or

magma: E := EllipticCurve("286650b1");

sage: E = EllipticCurve([1, -1, 0, -492, 566]) # or

sage: E = EllipticCurve("286650b1")

gp: E = ellinit([1, -1, 0, -492, 566]) \\ or

gp: E = ellinit("286650b1")

$$y^2 + x y = x^{3} - x^{2} - 492 x + 566$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-1, -32\right)$$ $$\left(25, -71\right)$$ $$\hat{h}(P)$$ ≈ 0.786669444069 1.1308210102

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-11, 73\right)$$, $$\left(-11, -62\right)$$, $$\left(-1, 33\right)$$, $$\left(-1, -32\right)$$, $$\left(25, 46\right)$$, $$\left(25, -71\right)$$, $$\left(59, 388\right)$$, $$\left(59, -447\right)$$, $$\left(83, 684\right)$$, $$\left(83, -767\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$286650$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$7546061250$$ = $$2 \cdot 3^{6} \cdot 5^{4} \cdot 7^{2} \cdot 13^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{590625}{338}$$ = $$2^{-1} \cdot 3^{3} \cdot 5^{5} \cdot 7 \cdot 13^{-2}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.741084683953$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$1.12977468111$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$12$$  = $$1\cdot2\cdot3\cdot1\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 286650.2.a.b

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{2} + q^{4} - q^{8} - 6q^{11} - q^{13} + q^{16} + q^{17} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 193536 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L^{(2)}(E,1)/2!$$ ≈ $$10.0471045499$$

## Local data

This elliptic curve is not semistable.

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

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
$$3$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$5$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$1$$ $$II$$ Additive -1 2 2 0
$$13$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 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 X5.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 0 & 1 \\ 1 & 1 \end{array}\right)$ and has index 2.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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 286650b 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 3.3.9800.1 $$\Z/2\Z$$ Not in database
6 6.6.768320000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.