# Properties

 Label 2394b4 Conductor $2394$ Discriminant $-9.107\times 10^{18}$ j-invariant $$\frac{361682234074684125}{462672528510976}$$ 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, 0, 400773, 107342693])

gp: E = ellinit([1, -1, 0, 400773, 107342693])

magma: E := EllipticCurve([1, -1, 0, 400773, 107342693]);

## Simplified equation

 $$y^2+xy=x^3-x^2+400773x+107342693$$ y^2+xy=x^3-x^2+400773x+107342693 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z+400773xz^2+107342693z^3$$ y^2z+xyz=x^3-x^2z+400773xz^2+107342693z^3 (dehomogenize, simplify) $$y^2=x^3+6412365x+6876344718$$ y^2=x^3+6412365x+6876344718 (homogenize, minimize)

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-202, 4357\right)$$ (-202, 4357) $\hat{h}(P)$ ≈ $0.63850304143453576872216833550$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{941}{4}, \frac{941}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-202, 4357\right)$$, $$\left(-202, -4155\right)$$, $$\left(-107, 8005\right)$$, $$\left(-107, -7898\right)$$, $$\left(4187, 272086\right)$$, $$\left(4187, -276273\right)$$, $$\left(16822, 2174917\right)$$, $$\left(16822, -2191739\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2394$$ = $2 \cdot 3^{2} \cdot 7 \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-9106783378681540608$ = $-1 \cdot 2^{12} \cdot 3^{9} \cdot 7^{4} \cdot 19^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{361682234074684125}{462672528510976}$$ = $2^{-12} \cdot 3^{6} \cdot 5^{3} \cdot 7^{-4} \cdot 19^{-6} \cdot 71^{3} \cdot 223^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.3239237997651270378920174692\dots$ Stable Faltings height: $1.4999645832640447693455835415\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.63850304143453576872216833550\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.15520971436105648709490479262\dots$ 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: $96$  = $2\cdot2\cdot2^{2}\cdot( 2 \cdot 3 )$ 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: $L'(E,1)$ ≈ $2.3784449923132826766216916003$

## 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);

$$q - q^{2} + q^{4} + q^{7} - q^{8} - 6 q^{11} + 2 q^{13} - q^{14} + q^{16} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 55296 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12
$3$ $2$ $III^{*}$ Additive 1 2 9 0
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$19$ $6$ $I_{6}$ Split multiplicative -1 1 6 6

## 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 2.3.0.1
$3$ 3B.1.2 3.8.0.2

## $p$-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 add ss split ord ord ss split ord ord ord ord ord ord ord 7 - 1,3 2 1 3 1,1 2 1 1 1 1 1 1 1 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, 3 and 6.
Its isogeny class 2394b consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $3$ 3.1.1323.1 $$\Z/6\Z$$ Not in database $4$ 4.2.155952.2 $$\Z/4\Z$$ Not in database $6$ 6.0.5250987.1 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.67371264.1 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $8$ 8.0.24321026304.2 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $18$ 18.0.32280955105118098668547786909650858258432.1 $$\Z/2\Z \oplus \Z/18\Z$$ 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.