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([1,0,0,0]),K([-1,-1,0,0]),K([-5/2,-5/2,1/2,1/2]),K([3,0,0,0]),K([7,3/2,-2,-1/2])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-1,-1,0,0]),Polrev([-5/2,-5/2,1/2,1/2]),Polrev([3,0,0,0]),Polrev([7,3/2,-2,-1/2])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![-1,-1,0,0],K![-5/2,-5/2,1/2,1/2],K![3,0,0,0],K![7,3/2,-2,-1/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/2a^3-3/2a^2+3/2a+11/2)\) | = | \((-1/2a^3-3/2a^2+3/2a+11/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 5 \) | = | \(5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1/2a^3-5a^2-27/2a+24)\) | = | \((-1/2a^3-3/2a^2+3/2a+11/2)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1953125 \) | = | \(-5^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{102109}{6250} a^{3} - \frac{1063917}{3125} a^{2} + \frac{972919}{6250} a + \frac{8947842}{3125} \) | ||
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 | $\left(\frac{1}{2} a^{2} - \frac{3}{2} : -\frac{3}{2} a^{2} + \frac{13}{2} : 1\right)$ |
| Height | \(0.036476401579014739116427926804177207823\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.036476401579014739116427926804177207823 \) | ||
| Period: | \( 185.18644608100145421800016071188693286 \) | ||
| Tamagawa product: | \( 9 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.18379990985357 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/2a^3-3/2a^2+3/2a+11/2)\) | \(5\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
5.2-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.