Properties

Label 406560fg1
Conductor $406560$
Discriminant $1.250\times 10^{12}$
j-invariant \( \frac{601211584}{11025} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -8510, -300192])
 
gp: E = ellinit([0, 1, 0, -8510, -300192])
 
magma: E := EllipticCurve([0, 1, 0, -8510, -300192]);
 

\(y^2=x^3+x^2-8510x-300192\)  Toggle raw display

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z \times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(481, 10350\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $4.9271914160832280525629144986$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-59, 0\right) \), \( \left(-48, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-59, 0\right) \), \( \left(-48, 0\right) \), \( \left(106, 0\right) \), \((481,\pm 10350)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 406560 \)  =  \(2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(1250013441600 \)  =  \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{601211584}{11025} \)  =  \(2^{6} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 211^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(1.1159596867833287263091776732\dots\)
Stable Faltings height: \(-0.42956153989582920043041017651\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(4.9271914160832280525629144986\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.49761836771920806846465603291\dots\)
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: \( 64 \)  = \( 2\cdot2\cdot2\cdot2\cdot2^{2} \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 406560.2.a.fg

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);
 

\( q + q^{3} + q^{5} + q^{7} + q^{9} - 6q^{13} + q^{15} + 6q^{17} - 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 983040
\( \Gamma_0(N) \)-optimal: not computed* (one of 2 curves in this isogeny class which might be optimal)
Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 400000. The Manin constant is correct provided that this curve is optimal.

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);
 

\( L'(E,1) \) ≈ \( 9.8074437996457172036871978768009501382 \)

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

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\((j)_{-}\)
\(2\) \(2\) \(III\) Additive 1 5 6 0
\(3\) \(2\) \(I_{2}\) Split multiplicative -1 1 2 2
\(5\) \(2\) \(I_{2}\) Split multiplicative -1 1 2 2
\(7\) \(2\) \(I_{2}\) Split multiplicative -1 1 2 2
\(11\) \(4\) \(I_0^{*}\) Additive -1 2 6 0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $$ and has index 6.

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 \( p \) except those listed.

prime Image of Galois representation
\(2\) Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 406560fg consists of 4 curves linked by isogenies of degrees dividing 4.