Properties

Label 424830f2
Conductor $424830$
Discriminant $-7.617\times 10^{29}$
j-invariant \( \frac{584669638645799}{386547056640} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, 2184001172, -14826048911792])
 
gp: E = ellinit([1, 1, 0, 2184001172, -14826048911792])
 
magma: E := EllipticCurve([1, 1, 0, 2184001172, -14826048911792]);
 

\(y^2+xy=x^3+x^2+2184001172x-14826048911792\)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{1708177506343}{185695129}, \frac{6208731169631759047}{2530467522883}\right) \)
\(\hat{h}(P)\) ≈  $25.525251959630190322017199651$

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 424830 \)  =  \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-761682788085010617870330101760 \)  =  \(-1 \cdot 2^{33} \cdot 3^{2} \cdot 5 \cdot 7^{10} \cdot 17^{8} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{584669638645799}{386547056640} \)  =  \(2^{-33} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{2} \cdot 17 \cdot 8887^{3}\)
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);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(25.525251959630190322017199651\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.016180759109758828160690552821\)
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: \( 6 \)  = \( 1\cdot2\cdot1\cdot1\cdot3 \)
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 424830.2.a.f

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 824481504
\( \Gamma_0(N) \)-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 curve 424830f1 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);
 

\( L'(E,1) \) ≈ \( 2.4781077190480534939204353170918944594 \)

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\((j)_{-}\)
\(2\) \(1\) \(I_{33}\) Non-split multiplicative 1 1 33 33
\(3\) \(2\) \(I_{2}\) Non-split multiplicative 1 1 2 2
\(5\) \(1\) \(I_{1}\) Non-split multiplicative 1 1 1 1
\(7\) \(1\) \(II^{*}\) Additive -1 2 10 0
\(17\) \(3\) \(IV^{*}\) Additive -1 2 8 0

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 \( p \) except those listed.

prime Image of Galois representation
\(3\) B

$p$-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 \(d=\) 3.
Its isogeny class 424830f consists of 2 curves linked by isogenies of degree 3.