# Properties

 Label 690.h3 Conductor $690$ Discriminant $3.554\times 10^{13}$ j-invariant $$\frac{71356102305927901489}{35540674560000}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -86411, -9808711])

gp: E = ellinit([1, 1, 1, -86411, -9808711])

magma: E := EllipticCurve([1, 1, 1, -86411, -9808711]);

$$y^2+xy+y=x^3+x^2-86411x-9808711$$

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-173, 86\right)$$, $$\left(339, -170\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-173, 86\right)$$, $$\left(339, -170\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$690$$ = $2 \cdot 3 \cdot 5 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $35540674560000$ = $2^{14} \cdot 3^{8} \cdot 5^{4} \cdot 23^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{71356102305927901489}{35540674560000}$$ = $2^{-14} \cdot 3^{-8} \cdot 5^{-4} \cdot 23^{-2} \cdot 4147729^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.5530565109182306802498826948\dots$ Stable Faltings height: $1.5530565109182306802498826948\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.27846476699936290427745904751\dots$ 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: $112$  = $( 2 \cdot 7 )\cdot2\cdot2\cdot2$ 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) 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); Special value: $L(E,1)$ ≈ $1.9492533689955403299422133325978271152$

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3584 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is semistable. There are 4 primes of bad reduction:

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$(j)_{-}$
$2$ $14$ $I_{14}$ Split multiplicative -1 1 14 14
$3$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$23$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.24.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 23 split nonsplit nonsplit ss nonsplit 2 0 0 2,2 0 0 0 0 0,0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 690.h consists of 2 curves linked by isogenies of degrees dividing 4.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-23})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{23})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.18339659776.1 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.4.1386741760000.11 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.495729936270000.3 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.