# Properties

 Label 714g2 Conductor $714$ Discriminant $4.477\times 10^{16}$ j-invariant $$\frac{82582985847542515777}{44772582831427584}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \times \Z/{4}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -90724, 2605541])

gp: E = ellinit([1, 1, 1, -90724, 2605541])

magma: E := EllipticCurve([1, 1, 1, -90724, 2605541]);

$$y^2+xy+y=x^3+x^2-90724x+2605541$$

## Mordell-Weil group structure

$$\Z/{2}\Z \times \Z/{4}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(29, -15\right)$$, $$\left(677, 15537\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-107, 3385\right)$$, $$\left(-107, -3279\right)$$, $$\left(29, -15\right)$$, $$\left(285, -143\right)$$, $$\left(677, 15537\right)$$, $$\left(677, -16215\right)$$

## 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: $$44772582831427584$$ = $$2^{12} \cdot 3^{8} \cdot 7^{8} \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{82582985847542515777}{44772582831427584}$$ = $$2^{-12} \cdot 3^{-8} \cdot 7^{-8} \cdot 13^{3} \cdot 17^{-2} \cdot 457^{3} \cdot 733^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.8859817887630724531175749997\dots$$ Stable Faltings height: $$1.8859817887630724531175749997\dots$$

## 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: $$0.31378491944261942823852023361\dots$$ 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: $$384$$  = $$( 2^{2} \cdot 3 )\cdot2\cdot2^{3}\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$8$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 7680 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$1.8827095166557165694311214016361321632$$

## 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$$(j)_{-}$$
$$2$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$3$$ $$2$$ $$I_{8}$$ Non-split multiplicative 1 1 8 8
$$7$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$17$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 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 X188c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 2 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 1 \end{array}\right)$ and has index 96.

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(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 1 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $4$ $$\Q(i, \sqrt{17})$$ $$\Z/4\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{17})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.5473632256.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.0.6179217664.3 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.4.2910381408256.21 $$\Z/2\Z \times \Z/16\Z$$ Not in database $8$ 8.2.35523982503387.4 $$\Z/2\Z \times \Z/12\Z$$ 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 $$\Z/2\Z \times \Z/24\Z$$ 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.