Properties

Label 422331ca1
Conductor $422331$
Discriminant $-2.913\times 10^{12}$
j-invariant \( -\frac{692224}{867} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -2760, 98375])
 
gp: E = ellinit([0, 1, 1, -2760, 98375])
 
magma: E := EllipticCurve([0, 1, 1, -2760, 98375]);
 

\(y^2+y=x^3+x^2-2760x+98375\)  Toggle raw display

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(-\frac{35123}{6084}, \frac{160095079}{474552}\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $11.594483907589775513663419121$

Integral points

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

\(\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 422331 \)  =  \(3 \cdot 7^{2} \cdot 13^{2} \cdot 17\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-2913270068163 \)  =  \(-1 \cdot 3 \cdot 7^{6} \cdot 13^{4} \cdot 17^{2} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{692224}{867} \)  =  \(-1 \cdot 2^{12} \cdot 3^{-1} \cdot 13^{2} \cdot 17^{-2}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(1.0850614219251787368997911275\dots\)
Stable Faltings height: \(-0.74287677175632349433738105808\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(11.594483907589775513663419121\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.72608801134704266017731599606\dots\)
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: \( 2 \)  = \( 1\cdot1\cdot1\cdot2 \)
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: \(1\) (exact)

Modular invariants

Modular form 422331.2.a.ca

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);
 

\( q + 2q^{2} + q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + q^{9} - 8q^{10} + 4q^{11} + 2q^{12} - 4q^{15} - 4q^{16} - q^{17} + 2q^{18} + 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1726560
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 16.837231526114296890650825318414656964 \)

Local data

This elliptic curve is not semistable. There are 4 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\((j)_{-}\)
\(3\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1
\(7\) \(1\) \(I_0^{*}\) Additive -1 2 6 0
\(13\) \(1\) \(IV\) Additive 1 2 4 0
\(17\) \(2\) \(I_{2}\) Non-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(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 no rational isogenies. Its isogeny class 422331ca consists of this curve only.