Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -197573, 33848381]); // or
magma: E := EllipticCurve("630i6");
sage: E = EllipticCurve([1, -1, 1, -197573, 33848381]) # or
sage: E = EllipticCurve("630i6")
gp: E = ellinit([1, -1, 1, -197573, 33848381]) \\ or
gp: E = ellinit("630i6")

$$y^2 + x y + y = x^{3} - x^{2} - 197573 x + 33848381$$

Mordell-Weil group structure

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

Torsion generators

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

$$\left(255, -128\right)$$, $$\left(495, 7312\right)$$

Integral points

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

$$\left(-513, 256\right)$$, $$\left(75, 4372\right)$$, $$\left(243, 256\right)$$, $$\left(255, -128\right)$$, $$\left(271, 256\right)$$, $$\left(495, 7312\right)$$

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$630$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$79042057113600$$ = $$2^{12} \cdot 3^{8} \cdot 5^{2} \cdot 7^{6}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{1169975873419524361}{108425318400}$$ = $$2^{-12} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-6} \cdot 19^{3} \cdot 31^{3} \cdot 1789^{3}$$ 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: $$0.583643541782$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$576$$  = $$( 2^{2} \cdot 3 )\cdot2^{2}\cdot2\cdot( 2 \cdot 3 )$$ 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 form630.2.a.h

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
4608 . This curve is not $$\Gamma_0(N)$$-optimal.

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)

$$L(E,1)$$ ≈ $$2.33457416713$$

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$$(j)_{-}$$
$$2$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$3$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$7$$ $$6$$ $$I_{6}$$ 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 7 split add nonsplit split 3 - 0 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 630.h 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 $$\Q(\sqrt{7}, \sqrt{15})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{3}, \sqrt{-5})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-3}, \sqrt{-7})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
6 6.0.110716875.2 $$\Z/6\Z \times \Z/6\Z$$ 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.