Properties

Label 407330.u1
Conductor 407330
Discriminant 9428469379581054687500
j-invariant \( \frac{1969902499564819009}{63690429687500} \)
CM no
Rank 1
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -13815375, 19203578125]); // or
 
magma: E := EllipticCurve("407330u4");
 
sage: E = EllipticCurve([1, 0, 0, -13815375, 19203578125]) # or
 
sage: E = EllipticCurve("407330u4")
 
gp: E = ellinit([1, 0, 0, -13815375, 19203578125]) \\ or
 
gp: E = ellinit("407330u4")
 

\( y^2 + x y = x^{3} - 13815375 x + 19203578125 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(5150, 288375\right) \)
\(\hat{h}(P)\) ≈  0.779013172367

Torsion generators

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

\( \left(\frac{7375}{4}, -\frac{7375}{8}\right) \)

Integral points

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

\( \left(-818, 173491\right) \), \( \left(-818, -172673\right) \), \( \left(750, 95875\right) \), \( \left(750, -96625\right) \), \( \left(5150, 288375\right) \), \( \left(5150, -293525\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 407330 \)  =  \(2 \cdot 5 \cdot 7 \cdot 11 \cdot 23^{2}\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(9428469379581054687500 \)  =  \(2^{2} \cdot 5^{12} \cdot 7^{2} \cdot 11^{3} \cdot 23^{6} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{1969902499564819009}{63690429687500} \)  =  \(2^{-2} \cdot 5^{-12} \cdot 7^{-2} \cdot 11^{-3} \cdot 23^{3} \cdot 54503^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.779013172366802\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.128811337570517\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 288 \)  = \( 2\cdot( 2^{2} \cdot 3 )\cdot2\cdot3\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 407330.2.a.u

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} - q^{7} + q^{8} + q^{9} + q^{10} + q^{11} - 2q^{12} - 4q^{13} - q^{14} - 2q^{15} + q^{16} + q^{18} + 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 43794432
\( \Gamma_0(N) \)-optimal: unknown* (one of 4 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 curve 407330.u3 is optimal.

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L'(E,1) \) ≈ \( 7.224892467668598 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(5\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(7\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(11\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(23\) \(2\) \( I_0^{*} \) 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 X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(2\) B
\(3\) B

$p$-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 non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 407330.u consists of 4 curves linked by isogenies of degrees dividing 6.