# Properties

 Label 690e2 Conductor $690$ Discriminant $49979073600$ j-invariant $$\frac{87109155423289}{49979073600}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -924, 922])

gp: E = ellinit([1, 0, 1, -924, 922])

magma: E := EllipticCurve([1, 0, 1, -924, 922]);

$$y^2+xy+y=x^3-924x+922$$

## 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);

 $P$ = $$\left(-4, 69\right)$$ $\hat{h}(P)$ ≈ $0.37469971446602829128269787869$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-31, 15\right)$$, $$\left(1, -1\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-31, 15\right)$$, $$\left(-22, 114\right)$$, $$\left(-22, -93\right)$$, $$\left(-7, 87\right)$$, $$\left(-7, -81\right)$$, $$\left(-4, 69\right)$$, $$\left(-4, -66\right)$$, $$\left(1, -1\right)$$, $$\left(41, 159\right)$$, $$\left(41, -201\right)$$, $$\left(50, 258\right)$$, $$\left(50, -309\right)$$, $$\left(116, 1149\right)$$, $$\left(116, -1266\right)$$, $$\left(185, 2391\right)$$, $$\left(185, -2577\right)$$, $$\left(2561, 128319\right)$$, $$\left(2561, -130881\right)$$

## 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: $49979073600$ = $2^{6} \cdot 3^{10} \cdot 5^{2} \cdot 23^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{87109155423289}{49979073600}$$ = $2^{-6} \cdot 3^{-10} \cdot 5^{-2} \cdot 23^{-2} \cdot 97^{3} \cdot 457^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.74256720306029198058528253412\dots$ Stable Faltings height: $0.74256720306029198058528253412\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.37469971446602829128269787869\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.96349517146160250299723501590\dots$ 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: $80$  = $2\cdot( 2 \cdot 5 )\cdot2\cdot2$ 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: $L'(E,1)$ ≈ $1.8051068281802971413506436546897045090$

## 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);

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - 6q^{13} - q^{15} + q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 960 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$3$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$23$ $2$ $I_{2}$ 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 8.12.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 nonsplit split nonsplit ss ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary 2 4 1 1,5 1 1 1 1 2 1 1 1 1 1 1 0 0 0 0,0 0 0 0 0 0 0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 690e 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 $K$ $4$ $$\Q(\sqrt{2}, \sqrt{-115})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{115})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.30983121016875.5 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ 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 $$\Z/2\Z \times \Z/8\Z$$ 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.