# Properties

 Label 2415.a4 Conductor $2415$ Discriminant $-53476250625$ j-invariant $$\frac{12152722588271}{53476250625}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 479, 10568])

gp: E = ellinit([1, 1, 1, 479, 10568])

magma: E := EllipticCurve([1, 1, 1, 479, 10568]);

$$y^2+xy+y=x^3+x^2+479x+10568$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-9, 79\right)$$ (-9, 79) $\hat{h}(P)$ ≈ $1.9148100889381927776664702123$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{61}{4}, \frac{57}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-9, 79\right)$$, $$\left(-9, -71\right)$$, $$\left(167, 2103\right)$$, $$\left(167, -2271\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2415$$ = $3 \cdot 5 \cdot 7 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-53476250625$ = $-1 \cdot 3^{12} \cdot 5^{4} \cdot 7 \cdot 23$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{12152722588271}{53476250625}$$ = $3^{-12} \cdot 5^{-4} \cdot 7^{-1} \cdot 23^{-1} \cdot 83^{3} \cdot 277^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.74102856999069735035248899634\dots$ Stable Faltings height: $0.74102856999069735035248899634\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.9148100889381927776664702123\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.80227915728347658564548664360\dots$ 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: $4$  = $2\cdot2\cdot1\cdot1$ 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: $1$ (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: $L'(E,1)$ ≈ $1.5362122245112321529414736512$

## 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);

$$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - q^{7} + 3 q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} - 6 q^{13} + q^{14} + q^{15} - q^{16} - 2 q^{17} - q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1920 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is 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$(j)_{-}$
$3$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$23$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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
$2$ 2B 4.6.0.1

## $p$-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 nonsplit nonsplit nonsplit ord ord ord ss nonsplit ord ss ord ord ord ss 4 3 1 3 1 1 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 0,0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 2415.a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-161})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{23})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-7})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-7}, \sqrt{23})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ 8.2.918400908166875.2 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.