# Properties

 Label 409101.k1 Conductor $409101$ Discriminant $-1.094\times 10^{17}$ j-invariant $$-\frac{1877953}{10412307}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -5629, 15911468]) # or

sage: E = EllipticCurve("409101k1")

gp: E = ellinit([1, 0, 0, -5629, 15911468]) \\ or

gp: E = ellinit("409101k1")

magma: E := EllipticCurve([1, 0, 0, -5629, 15911468]); // or

magma: E := EllipticCurve("409101k1");

$$y^2 + x y = x^{3} - 5629 x + 15911468$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(131, 4109\right)$$ $$\hat{h}(P)$$ ≈ $0.2928012978516078$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-232, 2294\right)$$, $$\left(-232, -2062\right)$$, $$\left(131, 4109\right)$$, $$\left(131, -4240\right)$$, $$\left(407, 8801\right)$$, $$\left(407, -9208\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$409101$$ = $$3 \cdot 7^{2} \cdot 11^{2} \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-109366553380274883$$ = $$-1 \cdot 3^{9} \cdot 7^{2} \cdot 11^{8} \cdot 23^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1877953}{10412307}$$ = $$-1 \cdot 3^{-9} \cdot 7 \cdot 11 \cdot 23^{-2} \cdot 29^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.292801297851608$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.267660910956651$$ 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: $$54$$  = $$3^{2}\cdot1\cdot3\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 409101.2.a.k

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);

$$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + 3q^{8} + q^{9} + 2q^{10} - q^{12} - 5q^{13} - 2q^{15} - q^{16} - q^{18} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3250368 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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);

$$L'(E,1)$$ ≈ $$4.232058954061557$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$3$$ $$9$$ $$I_{9}$$ Split multiplicative -1 1 9 9
$$7$$ $$1$$ $$II$$ Additive -1 2 2 0
$$11$$ $$3$$ $$IV^{*}$$ Additive -1 2 8 0
$$23$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ .

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 409101.k consists of this curve only.