# Properties

 Label 59290br2 Conductor $59290$ Discriminant $2.553\times 10^{16}$ j-invariant $$\frac{611960049}{122500}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -104869, -10545375]) # or

sage: E = EllipticCurve("59290br2")

gp: E = ellinit([1, -1, 0, -104869, -10545375]) \\ or

gp: E = ellinit("59290br2")

magma: E := EllipticCurve([1, -1, 0, -104869, -10545375]); // or

magma: E := EllipticCurve("59290br2");

$$y^2 + x y = x^{3} - x^{2} - 104869 x - 10545375$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-239, 1027\right)$$ $$\left(-201, 1644\right)$$ $$\hat{h}(P)$$ ≈ $2.0490936085263396$ $2.70217613074157$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-250, 125\right)$$, $$\left(366, -183\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-250, 125\right)$$, $$\left(-239, 1027\right)$$, $$\left(-239, -788\right)$$, $$\left(-201, 1644\right)$$, $$\left(-201, -1443\right)$$, $$\left(-129, 972\right)$$, $$\left(-129, -843\right)$$, $$\left(-124, 797\right)$$, $$\left(-124, -673\right)$$, $$\left(366, -183\right)$$, $$\left(436, 4927\right)$$, $$\left(436, -5363\right)$$, $$\left(641, 13292\right)$$, $$\left(641, -13933\right)$$, $$\left(1444, 52639\right)$$, $$\left(1444, -54083\right)$$, $$\left(4601, 308972\right)$$, $$\left(4601, -313573\right)$$, $$\left(7296, 618897\right)$$, $$\left(7296, -626193\right)$$, $$\left(108166, 35519917\right)$$, $$\left(108166, -35628083\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$59290$$ = $$2 \cdot 5 \cdot 7^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$25531741560902500$$ = $$2^{2} \cdot 5^{4} \cdot 7^{8} \cdot 11^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{611960049}{122500}$$ = $$2^{-2} \cdot 3^{3} \cdot 5^{-4} \cdot 7^{-2} \cdot 283^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.57331588796$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.269016327463$$ 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: $$128$$  = $$2\cdot2^{2}\cdot2^{2}\cdot2^{2}$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 59290.2.a.bc

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 491520 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$7.69024253635$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$7$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$11$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0

## 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 X38.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 6 & 7 \end{array}\right)$ and has index 12.

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(3,20) 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 ss split add add ordinary ordinary ss ss ordinary ordinary ordinary ordinary ordinary ordinary 5 2,2 3 - - 2 2 2,2 2,2 2 2 2 2 2 2 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.
Its isogeny class 59290br consists of 4 curves linked by isogenies of degrees dividing 4.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$4$ $$\Q(\sqrt{-7}, \sqrt{11})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{-11}, \sqrt{14})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{2}, \sqrt{77})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.