Properties

Label 409101.m1
Conductor $409101$
Discriminant $-3.560\times 10^{20}$
j-invariant \( -\frac{5862183923791}{4979799} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -15590429, 23709911490]) # or
 
sage: E = EllipticCurve("409101m1")
 
gp: E = ellinit([1, 0, 0, -15590429, 23709911490]) \\ or
 
gp: E = ellinit("409101m1")
 
magma: E := EllipticCurve([1, 0, 0, -15590429, 23709911490]); // or
 
magma: E := EllipticCurve("409101m1");
 

\( y^2 + x y = x^{3} - 15590429 x + 23709911490 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(4267, 184630\right) \)
\(\hat{h}(P)\) ≈  $0.5118910991245599$

Integral points

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

\( \left(2386, 8608\right) \), \( \left(2386, -10994\right) \), \( \left(4267, 184630\right) \), \( \left(4267, -188897\right) \)

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: \(-356000235061073984073 \)  =  \(-1 \cdot 3^{9} \cdot 7^{9} \cdot 11^{7} \cdot 23 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{5862183923791}{4979799} \)  =  \(-1 \cdot 3^{-9} \cdot 11^{-1} \cdot 13^{3} \cdot 19^{3} \cdot 23^{-1} \cdot 73^{3}\)
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: \(0.51189109912456\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.169031518124315\)
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: \( 72 \)  = \( 3^{2}\cdot2\cdot2^{2}\cdot1 \)
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: \(1\) (exact)

Modular invariants

Modular form 409101.2.a.m

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);
 

\( q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + 3q^{8} + q^{9} + 2q^{10} - q^{12} + 7q^{13} - 2q^{15} - q^{16} + 3q^{17} - q^{18} - q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 23224320
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 6.229852531153099 \)

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
\(7\) \(2\) \( III^{*} \) Additive -1 2 9 0
\(11\) \(4\) \( I_1^{*} \) 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.m consists of this curve only.