# Properties

 Label 2475g2 Conductor $2475$ Discriminant $1.005\times 10^{12}$ j-invariant $$\frac{169112377}{88209}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-x^2-2592x-15309$$ y^2+xy=x^3-x^2-2592x-15309 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-2592xz^2-15309z^3$$ y^2z+xyz=x^3-x^2z-2592xz^2-15309z^3 (dehomogenize, simplify) $$y^2=x^3-41475x-1021250$$ y^2=x^3-41475x-1021250 (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -2592, -15309])

gp: E = ellinit([1, -1, 0, -2592, -15309])

magma: E := EllipticCurve([1, -1, 0, -2592, -15309]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-6, 3\right)$$, $$\left(54, -27\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-6, 3\right)$$, $$\left(54, -27\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2475$$ = $3^{2} \cdot 5^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $1004755640625$ = $3^{12} \cdot 5^{6} \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{169112377}{88209}$$ = $3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 79^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.99481944315444378004694010414\dots$ Stable Faltings height: $-0.35920565739666125295106218093\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.70866128566071706184195204604\dots$ 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: $32$  = $2^{2}\cdot2^{2}\cdot2$ 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: $L(E,1)$ ≈ $1.4173225713214341236839040921$

## 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);

$$q + q^{2} - q^{4} - 4 q^{7} - 3 q^{8} - q^{11} + 2 q^{13} - 4 q^{14} - q^{16} - 2 q^{17} + O(q^{20})$$

 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 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$(j)_{-}$
$3$ $4$ $I_{6}^{*}$ Additive -1 2 12 6
$5$ $4$ $I_0^{*}$ Additive 1 2 6 0
$11$ $2$ $I_{2}$ Non-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 2.6.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 11 ord add add nonsplit 5 - - 0 1 - - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 2475g 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 $K$ $4$ $$\Q(\sqrt{3}, \sqrt{5})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-5}, \sqrt{33})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-11}, \sqrt{-15})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.20012416875.4 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ 16.0.36004060626969600000000.1 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ 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.