Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -33777, 2136861]); // or

magma: E := EllipticCurve("286650o1");

sage: E = EllipticCurve([1, -1, 0, -33777, 2136861]) # or

sage: E = EllipticCurve("286650o1")

gp: E = ellinit([1, -1, 0, -33777, 2136861]) \\ or

gp: E = ellinit("286650o1")

$$y^2 + x y = x^{3} - x^{2} - 33777 x + 2136861$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(25, 1131\right)$$ $$\hat{h}(P)$$ ≈ 5.12797018207

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(25, 1131\right)$$, $$\left(25, -1156\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) 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: $$509297558515200$$ = $$2^{9} \cdot 3^{7} \cdot 5^{2} \cdot 7^{2} \cdot 13^{5}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{4772330903305}{570306048}$$ = $$2^{-9} \cdot 3^{-1} \cdot 5 \cdot 7 \cdot 13^{-5} \cdot 5147^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$5.12797018207$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$0.504740687811$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$2$$  = $$1\cdot2\cdot1\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 286650.2.a.o

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q - q^{2} + q^{4} - q^{8} - 5q^{11} - q^{13} + q^{16} + 2q^{17} - 2q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1762560 $$\Gamma_0(N)$$-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/factorial(ar)

$$L'(E,1)$$ ≈ $$5.17659039354$$

## Local data

This elliptic curve is not semistable.

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

sage: E.local_data()

gp: ellglobalred(E)

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$3$$ $$2$$ $$I_1^{*}$$ Additive -1 2 7 1
$$5$$ $$1$$ $$II$$ Additive 1 2 2 0
$$7$$ $$1$$ $$II$$ Additive -1 2 2 0
$$13$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5

## Galois representations

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

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 286650o 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 3.3.382200.1 $$\Z/2\Z$$ Not in database
6 6.6.45575974080000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.