# Properties

 Label 714g3 Conductor $714$ Discriminant $5.525\times 10^{17}$ j-invariant $$\frac{70108386184777836280897}{552468975892674624}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -859044, -304722459])

gp: E = ellinit([1, 1, 1, -859044, -304722459])

magma: E := EllipticCurve([1, 1, 1, -859044, -304722459]);

$$y^2+xy+y=x^3+x^2-859044x-304722459$$

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-499, 249\right)$$, $$\left(1069, -535\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-499, 249\right)$$, $$\left(1069, -535\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$714$$ = $2 \cdot 3 \cdot 7 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $552468975892674624$ = $2^{6} \cdot 3^{16} \cdot 7^{4} \cdot 17^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{70108386184777836280897}{552468975892674624}$$ = $2^{-6} \cdot 3^{-16} \cdot 7^{-4} \cdot 17^{-4} \cdot 41234113^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.2325553790430451078261910604\dots$ Stable Faltings height: $2.2325553790430451078261910604\dots$

## 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: $0.15689245972130971411926011680\dots$ 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: $192$  = $( 2 \cdot 3 )\cdot2\cdot2^{2}\cdot2^{2}$ 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) 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: $L(E,1)$ ≈ $1.8827095166557165694311214016361321632$

## 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);

$$q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} + 2 q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 15360 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$3$ $2$ $I_{16}$ Non-split multiplicative 1 1 16 16
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$17$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## 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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.96.0.50

## $p$-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 7 17 split nonsplit split split 3 4 3 1 2 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 714g consists of 3 curves linked by isogenies of degrees dividing 8.

## 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 $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\zeta_{8})$$ $$\Z/4\Z \times \Z/4\Z$$ Not in database $4$ 4.2.100352.1 $$\Z/2\Z \times \Z/8\Z$$ Not in database $4$ $$\Q(i, \sqrt{17})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.841100226985984.69 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.5473632256.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.0.40282095616.10 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.2.35523982503387.4 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.