Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -8225100, 9078652800]); // or
magma: E := EllipticCurve("12870p6");
sage: E = EllipticCurve([1, -1, 0, -8225100, 9078652800]) # or
sage: E = EllipticCurve("12870p6")
gp: E = ellinit([1, -1, 0, -8225100, 9078652800]) \\ or
gp: E = ellinit("12870p6")

$$y^2 + x y = x^{3} - x^{2} - 8225100 x + 9078652800$$

Mordell-Weil group structure

$$\Z/{2}\Z \times \Z/{6}\Z$$

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(1632, -816\right)$$, $$\left(3660, 165480\right)$$

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(228, 84828\right)$$, $$\left(1515, 8895\right)$$, $$\left(1632, -816\right)$$, $$\left(1680, -840\right)$$, $$\left(1801, 9324\right)$$, $$\left(3660, 165480\right)$$

Note: only one of each pair $\pm P$ is listed.

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$12870$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$22441209177531315600$$ = $$2^{4} \cdot 3^{8} \cdot 5^{2} \cdot 11^{6} \cdot 13^{6}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{84415028961834287121601}{30783551683856400}$$ = $$2^{-4} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{6} \cdot 11^{-6} \cdot 13^{-6} \cdot 31^{3} \cdot 28879^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$0.210333969903$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$576$$  = $$2\cdot2^{2}\cdot2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$12$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 12870.2.a.c

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 663552 $$\Gamma_0(N)$$-optimal: no 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)

$$L(E,1)$$ ≈ $$0.841335879614$$

Local data

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$11$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$13$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6

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 X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  and has index 6.

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 $$p$$ except those listed.

prime Image of Galois representation
$$2$$ Cs
$$3$$ B.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 11 13 nonsplit add nonsplit split split 7 - 0 1 1 1 - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 12870p consists of 8 curves linked by isogenies of degrees dividing 12.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{-13}, \sqrt{-165})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{13})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{3}, \sqrt{55})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.1771470000.3 $$\Z/6\Z \times \Z/6\Z$$ 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.