Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, 153413, -149478166]); // or

magma: E := EllipticCurve("164080j1");

sage: E = EllipticCurve([0, 0, 0, 153413, -149478166]) # or

sage: E = EllipticCurve("164080j1")

gp: E = ellinit([0, 0, 0, 153413, -149478166]) \\ or

gp: E = ellinit("164080j1")

$$y^2 = x^{3} + 153413 x - 149478166$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(\frac{5077}{9}, \frac{291520}{27}\right)$$ $$\hat{h}(P)$$ ≈ 5.09085271509

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$164080$$ = $$2^{4} \cdot 5 \cdot 7 \cdot 293$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-9883570135040000000$$ = $$-1 \cdot 2^{19} \cdot 5^{7} \cdot 7^{7} \cdot 293$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{97487796040631919}{2412980990000000}$$ = $$2^{-7} \cdot 3^{3} \cdot 5^{-7} \cdot 7^{-7} \cdot 13^{3} \cdot 293^{-1} \cdot 11801^{3}$$ 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: $$5.09085271509$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$0.111087356448$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$28$$  = $$2^{2}\cdot7\cdot1\cdot1$$ 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 164080.2.a.n

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q + 3q^{3} + q^{5} - q^{7} + 6q^{9} - 5q^{11} + 7q^{13} + 3q^{15} - 3q^{17} + 8q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 9483264 $$\Gamma_0(N)$$-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)

$$L'(E,1)$$ ≈ $$15.8348223652$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_11^{*}$$ Additive -1 4 19 7
$$5$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$7$$ $$1$$ $$I_{7}$$ Non-split multiplicative 1 1 7 7
$$293$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## 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 $$p$$ except those listed.

prime Image of Galois representation
$$7$$ B.6.1

## $p$-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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 164080j consists of 2 curves linked by isogenies of degree 7.

## 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
2 $$\Q(\sqrt{-1})$$ $$\Z/7\Z$$ Not in database
3 3.1.82040.1 $$\Z/2\Z$$ Not in database
6 $$x^{6} - 78 x^{4} + 1521 x^{2} + 82040$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$x^{6} - 23 x^{4} - 116 x^{3} + 172 x^{2} + 1856 x + 3560$$ $$\Z/14\Z$$ 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.