Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -156559, -15923518]); // or
magma: E := EllipticCurve("2730n6");
sage: E = EllipticCurve([1, 0, 1, -156559, -15923518]) # or
sage: E = EllipticCurve("2730n6")
gp: E = ellinit([1, 0, 1, -156559, -15923518]) \\ or
gp: E = ellinit("2730n6")

$$y^2 + x y + y = x^{3} - 156559 x - 15923518$$

## Mordell-Weil group structure

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

## Torsion generators

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

$$\left(-329, 164\right)$$, $$\left(439, -220\right)$$

## Integral points

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

$$\left(-329, 164\right)$$, $$\left(439, -220\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$2730$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$136231857216000000$$ = $$2^{12} \cdot 3^{2} \cdot 5^{6} \cdot 7^{2} \cdot 13^{6}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{424378956393532177129}{136231857216000000}$$ = $$2^{-12} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-2} \cdot 13^{-6} \cdot 43^{3} \cdot 174763^{3}$$ 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: $$0.246113246022$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$96$$  = $$2\cdot2\cdot2\cdot2\cdot( 2 \cdot 3 )$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$4$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form2730.2.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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 41472 $$\Gamma_0(N)$$-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)

$$L(E,1)$$ ≈ $$1.47667947613$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$13$$ $$6$$ $$I_{6}$$ 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.1.2

## $p$-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 13 nonsplit split nonsplit split split 3 1 0 1 1 0 1 0 0 0

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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 2730n 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-3})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
3 3.1.11907.2 $$\Z/2\Z \times \Z/6\Z$$ Not in database
4 $$\Q(\sqrt{-3}, \sqrt{-35})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
$$\Q(\sqrt{-13}, \sqrt{35})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{3}, \sqrt{13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
6 6.0.425329947.5 $$\Z/6\Z \times \Z/6\Z$$ 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.