# Properties

 Label 86640.di3 Conductor $86640$ Discriminant $6.261\times 10^{16}$ j-invariant $$\frac{758800078561}{324900}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -1097560, 442049300]) # or

sage: E = EllipticCurve("86640eh2")

gp: E = ellinit([0, 1, 0, -1097560, 442049300]) \\ or

gp: E = ellinit("86640eh2")

magma: E := EllipticCurve([0, 1, 0, -1097560, 442049300]); // or

magma: E := EllipticCurve("86640eh2");

$$y^2 = x^{3} + x^{2} - 1097560 x + 442049300$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(1964, 76590\right)$$ $$\hat{h}(P)$$ ≈ $6.238172410793409$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(595, 0\right)$$, $$\left(614, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1210, 0\right)$$, $$\left(595, 0\right)$$, $$\left(614, 0\right)$$, $$(1964,\pm 76590)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$86640$$ = $$2^{4} \cdot 3 \cdot 5 \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$62608206794342400$$ = $$2^{14} \cdot 3^{2} \cdot 5^{2} \cdot 19^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{758800078561}{324900}$$ = $$2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{3} \cdot 19^{-2} \cdot 1303^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$6.23817241079$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.344237997707$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$64$$  = $$2^{2}\cdot2\cdot2\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 86640.2.a.di

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{3} + q^{5} - 4q^{7} + q^{9} + 4q^{11} + 2q^{13} + q^{15} - 2q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1658880 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$8.58966392018$$

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_6^{*}$$ Additive -1 4 14 2
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$19$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2

## 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.

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

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

## $p$-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.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add split split ordinary ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ss - 2 2 1 3 1 1 - 1 1 1 1 1 1 1,1 - 0 0 0 0 0 0 - 0 0 0 0 0 0 0,0

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.
Its isogeny class 86640.di consists of 4 curves linked by isogenies of degrees dividing 4.

## 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
$4$ $$\Q(\sqrt{5}, \sqrt{-19})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{6}, \sqrt{19})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{-5}, \sqrt{-114})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.