# Properties

 Label 5712.q2 Conductor $5712$ Discriminant $5.914\times 10^{13}$ j-invariant $$\frac{34623662831857}{14438442312}$$ CM no Rank $1$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -10864, 226772])

gp: E = ellinit([0, 1, 0, -10864, 226772])

magma: E := EllipticCurve([0, 1, 0, -10864, 226772]);

$$y^2=x^3+x^2-10864x+226772$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(98, 336\right)$$ $\hat{h}(P)$ ≈ $0.92529736548905622550501741134$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-28, 714\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-62,\pm 816)$$, $$(-28,\pm 714)$$, $$\left(91, 0\right)$$, $$(98,\pm 336)$$, $$(532,\pm 12054)$$, $$(2114,\pm 97104)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$5712$$ = $2^{4} \cdot 3 \cdot 7 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $59139859709952$ = $2^{15} \cdot 3^{2} \cdot 7^{4} \cdot 17^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{34623662831857}{14438442312}$$ = $2^{-3} \cdot 3^{-2} \cdot 7^{-4} \cdot 11^{3} \cdot 17^{-4} \cdot 2963^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.3396970510326073729443042822\dots$ Stable Faltings height: $0.64654987047266206352707216074\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.92529736548905622550501741134\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.56552859261187456988327462876\dots$ 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: $128$  = $2^{2}\cdot2\cdot2^{2}\cdot2^{2}$ 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: $L'(E,1)$ ≈ $4.1862569348200102891314433981103487565$

## 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);

$$q + q^{3} - 2q^{5} + q^{7} + q^{9} - 6q^{13} - 2q^{15} + q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 18432 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not 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$(j)_{-}$
$2$ $4$ $I_7^{*}$ Additive -1 4 15 3
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $4$ $I_{4}$ 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
$2$ 2B 8.24.0.49

## $p$-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 add split ordinary split ss ordinary split ss ordinary ordinary ordinary ordinary ordinary ordinary ss - 2 3 2 1,1 1 2 3,1 1 1 1 1 3 1 1,1 - 0 0 0 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 and 4.
Its isogeny class 5712.q consists of 3 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ 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 $4$ 4.0.1019592.2 $$\Z/8\Z$$ Not in database $8$ 8.0.1358954496.9 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.66532342173696.5 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.8.3364400907943936.14 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ 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.