# Properties

 Label 409101bb1 Conductor $409101$ Discriminant $-850520979$ j-invariant $$-\frac{134217728}{1863}$$ 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, -821, 8894]) # or

sage: E = EllipticCurve("409101bb1")

gp: E = ellinit([0, 1, 1, -821, 8894]) \\ or

gp: E = ellinit("409101bb1")

magma: E := EllipticCurve([0, 1, 1, -821, 8894]); // or

magma: E := EllipticCurve("409101bb1");

$$y^2 + y = x^{3} + x^{2} - 821 x + 8894$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(16, 10\right)$$ $$\left(-26, 115\right)$$ $$\hat{h}(P)$$ ≈ $0.6971746600317436$ $0.8076868690931983$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-32, 58\right)$$, $$\left(-32, -59\right)$$, $$\left(-26, 115\right)$$, $$\left(-26, -116\right)$$, $$\left(16, 10\right)$$, $$\left(16, -11\right)$$, $$\left(18, 16\right)$$, $$\left(18, -17\right)$$, $$\left(76, 625\right)$$, $$\left(76, -626\right)$$, $$\left(898, 26911\right)$$, $$\left(898, -26912\right)$$

## 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: $$-850520979$$ = $$-1 \cdot 3^{4} \cdot 7^{3} \cdot 11^{3} \cdot 23$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{134217728}{1863}$$ = $$-1 \cdot 2^{27} \cdot 3^{-4} \cdot 23^{-1}$$ 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: $$0.560357089061539$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$1.58769012385216$$ 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: $$16$$  = $$2^{2}\cdot2\cdot2\cdot1$$ 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.bb

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);

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

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 141312 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$14.234774658136805$$

## 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$$(j)_{-}$$
$$3$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$7$$ $$2$$ $$III$$ Additive -1 2 3 0
$$11$$ $$2$$ $$III$$ Additive 1 2 3 0
$$23$$ $$1$$ $$I_{1}$$ 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 409101bb consists of this curve only.