Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, 2514857562, -51791164403912]); // or

magma: E := EllipticCurve("86190bm4");

sage: E = EllipticCurve([1, 0, 1, 2514857562, -51791164403912]) # or

sage: E = EllipticCurve("86190bm4")

gp: E = ellinit([1, 0, 1, 2514857562, -51791164403912]) \\ or

gp: E = ellinit("86190bm4")

$$y^2 + x y + y = x^{3} + 2514857562 x - 51791164403912$$

## Mordell-Weil group structure

$$\Z/{4}\Z$$

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(77429, 24601485\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(77429, 24601485\right)$$

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

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$86190$$ = $$2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-2176708896205201761252525000000$$ = $$-1 \cdot 2^{6} \cdot 3^{24} \cdot 5^{8} \cdot 13^{7} \cdot 17^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{364421318680576777174674911}{450962301637624725000000}$$ = $$2^{-6} \cdot 3^{-24} \cdot 5^{-8} \cdot 11^{3} \cdot 13^{-1} \cdot 17^{-3} \cdot 109^{3} \cdot 191^{3} \cdot 3119^{3}$$ 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: $$0.013941557852$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$4608$$  = $$2\cdot( 2^{3} \cdot 3 )\cdot2^{3}\cdot2^{2}\cdot3$$ 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 form 86190.2.a.bm

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 185794560 $$\Gamma_0(N)$$-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)

$$L(E,1)$$ ≈ $$4.01516866138$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$24$$ $$I_{24}$$ Split multiplicative -1 1 24 24
$$5$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$13$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1
$$17$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3

## 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 X13h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 3 \end{array}\right)$ and has index 12.

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
$$2$$ B
$$3$$ B

## $p$-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 13 17 nonsplit split split add split 4 7 1 - 3 2 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, 4, 6 and 12.
Its isogeny class 86190.bm 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-221})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{13})$$ $$\Z/12\Z$$ Not in database
4 $$\Q(\sqrt{13}, \sqrt{-17})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database
4.2.2390336.1 $$\Z/24\Z$$ Not in database
6 6.0.6265569375.1 $$\Z/12\Z$$ 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.