Base field 4.4.12400.1
Generator \(a\), with minimal polynomial \( x^{4} - 12 x^{2} + 31 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([31, 0, -12, 0, 1]))
gp: K = nfinit(Polrev([31, 0, -12, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 0, -12, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0]),K([5/2,9/2,-1/2,-1/2]),K([0,0,0,0]),K([-78,64,7,-8]),K([604,-631/2,-72,79/2])])
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([5/2,9/2,-1/2,-1/2]),Polrev([0,0,0,0]),Polrev([-78,64,7,-8]),Polrev([604,-631/2,-72,79/2])], K);
magma: E := EllipticCurve([K![0,1,0,0],K![5/2,9/2,-1/2,-1/2],K![0,0,0,0],K![-78,64,7,-8],K![604,-631/2,-72,79/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^2+a-1/2)\) | = | \((1/2a^2+a-1/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1/2a^3+2a^2-3/2a-19)\) | = | \((1/2a^2+a-1/2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6561 \) | = | \(9^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{8025637785355}{81} a^{3} + \frac{46061580093845}{162} a^{2} - \frac{30209462949730}{81} a - \frac{57788918959565}{54} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(a^{3} - \frac{1}{2} a^{2} - 7 a + \frac{11}{2} : \frac{1}{2} a^{3} - \frac{9}{2} a^{2} - \frac{13}{2} a + \frac{57}{2} : 1\right)$ |
| Height | \(0.15320017629013892723823349302744101440\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{9}{2} a + \frac{3}{2} : -\frac{1}{8} a^{3} - \frac{3}{4} a^{2} - \frac{3}{4} a + \frac{31}{4} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.15320017629013892723823349302744101440 \) | ||
| Period: | \( 525.22095584861521324481889691911430445 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.89035015793608 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^2+a-1/2)\) | \(9\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.