Properties

Label 231280.g1
Conductor \(231280\)
Discriminant \(-719966224262994329600\)
j-invariant \( -\frac{581453267835321}{73208248217600} \)
CM no
Rank \(1\)
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -372547, -1293927614]); // or
magma: E := EllipticCurve("231280g1");
sage: E = EllipticCurve([0, 0, 0, -372547, -1293927614]) # or
sage: E = EllipticCurve("231280g1")
gp: E = ellinit([0, 0, 0, -372547, -1293927614]) \\ or
gp: E = ellinit("231280g1")

\( y^2 = x^{3} - 372547 x - 1293927614 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(1505, 39424\right) \)
\(\hat{h}(P)\) ≈  2.96979343254

Integral points

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

\( \left(1505, 39424\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 231280 \)  =  \(2^{4} \cdot 5 \cdot 7^{2} \cdot 59\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(-719966224262994329600 \)  =  \(-1 \cdot 2^{24} \cdot 5^{2} \cdot 7^{4} \cdot 59^{5} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( -\frac{581453267835321}{73208248217600} \)  =  \(-1 \cdot 2^{-12} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{2} \cdot 59^{-5} \cdot 7603^{3}\)
\( \text{End} (E) \)  =  \(\Z\)   (no Complex Multiplication)
\( \text{ST} (E) \)  =  $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(1\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  ≈  \(2.96979343254\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(0.0712275730201\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 24 \)  = \( 2^{2}\cdot2\cdot3\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 231280.2.1.g

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)

\( q - 3q^{3} + q^{5} + 6q^{9} + 2q^{11} - 6q^{13} - 3q^{15} - q^{17} + q^{19} + O(q^{20}) \)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
14653440 : curve is \( \Gamma_0(N) \)-optimal

Special L-value attached to the curve

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])

\( L'(E,1) \) ≈ \( 5.07674828569 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_16^{*} \) Additive -1 4 24 12
\(5\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(7\) \(3\) \( IV \) Additive 1 2 4 0
\(59\) \(1\) \( I_{5} \) Non-split multiplicative 1 1 5 5

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 \( p \) .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

No \(p\)-adic data exists for this curve.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has no rational isogenies. Its isogeny class 231280.g consists of this curve only.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2891.3 \(\Z/2\Z\) Not in database
6 6.0.493114979.2 \(\Z/2\Z \times \Z/2\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.