# Properties

 Label 3136s2 Conductor $3136$ Discriminant $377801998336$ j-invariant $$\frac{740772}{49}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-3724x+82320$$ y^2=x^3-3724x+82320 (homogenize, simplify) $$y^2z=x^3-3724xz^2+82320z^3$$ y^2z=x^3-3724xz^2+82320z^3 (dehomogenize, simplify) $$y^2=x^3-3724x+82320$$ y^2=x^3-3724x+82320 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -3724, 82320])

gp: E = ellinit([0, 0, 0, -3724, 82320])

magma: E := EllipticCurve([0, 0, 0, -3724, 82320]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(92, 720\right)$$ (92, 720) $\hat{h}(P)$ ≈ $3.8661144666795749330333708156$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(28, 0\right)$$, $$\left(42, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-70, 0\right)$$, $$\left(28, 0\right)$$, $$\left(42, 0\right)$$, $$(92,\pm 720)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$3136$$ = $2^{6} \cdot 7^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $377801998336$ = $2^{16} \cdot 7^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{740772}{49}$$ = $2^{2} \cdot 3^{3} \cdot 7^{-2} \cdot 19^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.97092224772906916420613283473\dots$ Stable Faltings height: $-0.92622906754518123423618636560\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $3.8661144666795749330333708156\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.93493147424956756866377454482\dots$ 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: $16$  = $2^{2}\cdot2^{2}$ 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: $L'(E,1)$ ≈ $3.6145520979503156654594075894$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3072 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$2$ $4$ $I_{6}^{*}$ Additive -1 6 16 0
$7$ $4$ $I_{2}^{*}$ Additive -1 2 8 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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.24.0.7
sage: gens = [[27, 54, 0, 55], [39, 52, 22, 47], [53, 4, 52, 5], [1, 4, 0, 1], [1, 0, 4, 1], [41, 52, 0, 55]]

sage: GL(2,Integers(56)).subgroup(gens)

magma: Gens := [[27, 54, 0, 55], [39, 52, 22, 47], [53, 4, 52, 5], [1, 4, 0, 1], [1, 0, 4, 1], [41, 52, 0, 55]];

magma: sub<GL(2,Integers(56))|Gens>;

The image of the adelic Galois representation has level $56$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 27 & 54 \\ 0 & 55 \end{array}\right),\left(\begin{array}{rr} 39 & 52 \\ 22 & 47 \end{array}\right),\left(\begin{array}{rr} 53 & 4 \\ 52 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 52 \\ 0 & 55 \end{array}\right)$

## $p$-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 ss ord add ord ord ord ord ss ord ord ord ord ord ord - 7,3 1 - 1 1 1 1 1,1 1 1 1 1 1 1 - 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.
Its isogeny class 3136s consists of 4 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{-2}, \sqrt{-7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.157351936.1 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.16862305517568.25 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.59447875862838378496.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ 16.8.1622647227216566419456.1 $$\Z/2\Z \oplus \Z/8\Z$$ 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.