Properties

Label 409101bf1
Conductor $409101$
Discriminant $-3.986\times 10^{17}$
j-invariant \( \frac{1605632000}{93710763} \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, 39527, -30211604]) # or
 
sage: E = EllipticCurve("409101bf1")
 
gp: E = ellinit([0, 1, 1, 39527, -30211604]) \\ or
 
gp: E = ellinit("409101bf1")
 
magma: E := EllipticCurve([0, 1, 1, 39527, -30211604]); // or
 
magma: E := EllipticCurve("409101bf1");
 

\( y^2 + y = x^{3} + x^{2} + 39527 x - 30211604 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(1052, 34303\right) \)\( \left(326, 4174\right) \)
\(\hat{h}(P)\) ≈  $0.6453903314176684$$1.3615541602260914$

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(296, 2740\right) \), \( \left(296, -2741\right) \), \( \left(326, 4174\right) \), \( \left(326, -4175\right) \), \( \left(1052, 34303\right) \), \( \left(1052, -34304\right) \), \( \left(1196, 41584\right) \), \( \left(1196, -41585\right) \), \( \left(1430, 54337\right) \), \( \left(1430, -54338\right) \), \( \left(25442, 4058338\right) \), \( \left(25442, -4058339\right) \), \( \left(275480, 144589252\right) \), \( \left(275480, -144589253\right) \)

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: \(-398600413559514243 \)  =  \(-1 \cdot 3^{11} \cdot 7^{4} \cdot 11^{6} \cdot 23^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1605632000}{93710763} \)  =  \(2^{18} \cdot 3^{-11} \cdot 5^{3} \cdot 7^{2} \cdot 23^{-2}\)
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: \(0.874636430574512\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.143235682704585\)
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: \( 132 \)  = \( 11\cdot3\cdot2\cdot2 \)
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: \(1\) (rounded)

Modular invariants

Modular form 409101.2.a.bf

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);
 

\( q + q^{3} - 2q^{4} + q^{9} - 2q^{12} + 3q^{13} + 4q^{16} - 2q^{17} - 3q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2956800
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 16.536847305216686 \)

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\((j)_{-}\)
\(3\) \(11\) \( I_{11} \) Split multiplicative -1 1 11 11
\(7\) \(3\) \( IV \) Additive 1 2 4 0
\(11\) \(2\) \( I_0^{*} \) Additive -1 2 6 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 409101bf consists of this curve only.