# Properties

 Label 93058.a1 Conductor $93058$ Discriminant $4.237\times 10^{16}$ j-invariant $$\frac{24553362849625}{1755162752}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -174996, 26364434])

gp: E = ellinit([1, 0, 1, -174996, 26364434])

magma: E := EllipticCurve([1, 0, 1, -174996, 26364434]);

$$y^2+xy+y=x^3-174996x+26364434$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(92, 3277\right)$$ $$\left(296, 574\right)$$ $\hat{h}(P)$ ≈ $0.65612520641594665208535405603$ $2.8202511129291088868470009348$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{759}{4}, -\frac{763}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-316, 7238\right)$$, $$\left(-316, -6923\right)$$, $$\left(92, 3277\right)$$, $$\left(92, -3370\right)$$, $$\left(296, 574\right)$$, $$\left(296, -871\right)$$, $$\left(322, 1690\right)$$, $$\left(322, -2013\right)$$, $$\left(552, 9625\right)$$, $$\left(552, -10178\right)$$, $$\left(874, 22827\right)$$, $$\left(874, -23702\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$93058$$ = $2 \cdot 7 \cdot 17^{2} \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $42365362032629888$ = $2^{7} \cdot 7^{2} \cdot 17^{6} \cdot 23^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{24553362849625}{1755162752}$$ = $2^{-7} \cdot 5^{3} \cdot 7^{-2} \cdot 23^{-4} \cdot 5813^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.9373602343101826667458522575\dots$ Stable Faltings height: $0.52075356228207462662108494856\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.8003209281366371368025476354\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.35415569443449929573816145797\dots$ 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: $32$  = $1\cdot2\cdot2^{2}\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (rounded) 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: $L^{(2)}(E,1)/2!$ ≈ $5.1007512680735442191081697559861269924$

## Modular invariants

Modular form 93058.2.a.a

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1032192 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $I_{7}$ Non-split multiplicative 1 1 7 7
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$17$ $4$ $I_0^{*}$ Additive 1 2 6 0
$23$ $4$ $I_{4}$ 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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.6.0.6

## $p$-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 nonsplit ordinary ss nonsplit ordinary ss add ordinary split ordinary ordinary ordinary ordinary ordinary ordinary 10 2 2,6 2 2 4,2 - 2 3 2 2 2 2 2 4 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 93058.a consists of 2 curves linked by isogenies of degree 2.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.0.453152.4 $$\Z/4\Z$$ Not in database $8$ 8.0.13142191046656.41 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.17165310754816.29 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.