# Properties

 Label 409101.c1 Conductor $409101$ Discriminant $-6.461\times 10^{20}$ j-invariant $$\frac{5319051554816}{3099832659}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, 2156180, -102986346]) # or

sage: E = EllipticCurve("409101c1")

gp: E = ellinit([0, -1, 1, 2156180, -102986346]) \\ or

gp: E = ellinit("409101c1")

magma: E := EllipticCurve([0, -1, 1, 2156180, -102986346]); // or

magma: E := EllipticCurve("409101c1");

$$y^2 + y = x^{3} - x^{2} + 2156180 x - 102986346$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(1489, 80041\right)$$ $$\left(279, 22808\right)$$ $$\hat{h}(P)$$ ≈ $1.2239207018206704$ $4.375839519092801$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(48, 786\right)$$, $$\left(48, -787\right)$$, $$\left(279, 22808\right)$$, $$\left(279, -22809\right)$$, $$\left(1489, 80041\right)$$, $$\left(1489, -80042\right)$$, $$\left(5458, 417406\right)$$, $$\left(5458, -417407\right)$$

## 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: $$-646074500666393526651$$ = $$-1 \cdot 3^{6} \cdot 7^{11} \cdot 11^{7} \cdot 23$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{5319051554816}{3099832659}$$ = $$2^{12} \cdot 3^{-6} \cdot 7^{-5} \cdot 11^{-1} \cdot 23^{-1} \cdot 1091^{3}$$ 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: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$5.30356840302763$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0956532541389279$$ 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: $$16$$  = $$2\cdot2\cdot2^{2}\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 409101.2.a.c

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);

$$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + q^{9} - 2q^{10} - 2q^{12} + 6q^{13} - q^{15} - 4q^{16} - 2q^{17} - 2q^{18} + 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 27648000 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$8.116857220767836$$

## 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$$(j)_{-}$$
$$3$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$7$$ $$2$$ $$I_5^{*}$$ Additive -1 2 11 5
$$11$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$23$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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.c consists of this curve only.