# Properties

 Label 585.g2 Conductor $585$ Discriminant $433128515625$ j-invariant $$\frac{59319456301170001}{594140625}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-73125x+7629336$$ y^2+xy=x^3-x^2-73125x+7629336 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-73125xz^2+7629336z^3$$ y^2z+xyz=x^3-x^2z-73125xz^2+7629336z^3 (dehomogenize, simplify) $$y^2=x^3-1170003x+487107502$$ y^2=x^3-1170003x+487107502 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -73125, 7629336])

gp: E = ellinit([1, -1, 0, -73125, 7629336])

magma: E := EllipticCurve([1, -1, 0, -73125, 7629336]);

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);

 $P$ = $$\left(120, 696\right)$$ (120, 696) $\hat{h}(P)$ ≈ $2.7724276067073725060266561064$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-312, 156\right)$$, $$\left(156, -78\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-312, 156\right)$$, $$\left(120, 696\right)$$, $$\left(120, -816\right)$$, $$\left(156, -78\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$585$$ = $3^{2} \cdot 5 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $433128515625$ = $3^{8} \cdot 5^{8} \cdot 13^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{59319456301170001}{594140625}$$ = $3^{-2} \cdot 5^{-8} \cdot 13^{-2} \cdot 390001^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.3921496218487160779632034313\dots$ Stable Faltings height: $0.84284347751466123226558081284\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.7724276067073725060266561064\dots$ 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: $0.85108145292620109686565595561\dots$ 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: $16$  = $2^{2}\cdot2\cdot2$ 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: $L'(E,1)$ ≈ $2.3595617156492210218470390653$

## 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);

$$q + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} - 4 q^{11} + q^{13} - q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1536 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$3$ $4$ $I_{2}^{*}$ Additive -1 2 8 2
$5$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8
$13$ $2$ $I_{2}$ Split multiplicative -1 1 2 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 8.48.0.108

The image of the adelic Galois representation has level $3120$, index $768$, and genus $13$.

## $p$-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 ord add nonsplit ss ord split ord ord ord ord ord ord ord ord ord 2 - 1 1,1 1 2 1 1 1 1 1 1 3 3 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, 4 and 8.
Its isogeny class 585.g consists of 8 curves linked by isogenies of degrees dividing 16.

## 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 $K$ $2$ $$\Q(\sqrt{3})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{13})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-13})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{26})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.592240896.1 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ $$\Q(\zeta_{24})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.56299900176.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.3162184666875.6 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.22986704741655040229376.1 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.16806995817636085862174923161600000000.71 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ 16.16.16806995817636085862174923161600000000.8 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ 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.