Base field 4.4.725.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 3 x^{2} + x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -3, -1, 1]))
gp: K = nfinit(Polrev([1, 1, -3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0,0]),K([1,-1,0,0]),K([-1,-1,1,0]),K([-12538,13737,8759,-6039]),K([-625284,685551,440180,-299268])])
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([1,-1,0,0]),Polrev([-1,-1,1,0]),Polrev([-12538,13737,8759,-6039]),Polrev([-625284,685551,440180,-299268])], K);
magma: E := EllipticCurve([K![0,0,0,0],K![1,-1,0,0],K![-1,-1,1,0],K![-12538,13737,8759,-6039],K![-625284,685551,440180,-299268]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^3+2a^2+4a-1)\) | = | \((-2a^3+2a^2+4a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-31250a^3+31250a^2+62500a-15625)\) | = | \((-2a^3+2a^2+4a-1)^{13}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1490116119384765625 \) | = | \(25^{13}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{9401666179616768}{78125} a^{3} + \frac{9401666179616768}{78125} a^{2} + \frac{18803332359233536}{78125} a - \frac{15212221580529664}{78125} \) | ||
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: | \(0\) |
| 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 0.090493666099753971099814291665807220455 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.567983715209496 \) | ||
| Analytic order of Ш: | \( 169 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^3+2a^2+4a-1)\) | \(25\) | \(1\) | \(I_{13}\) | Non-split multiplicative | \(1\) | \(1\) | \(13\) | \(13\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(13\) | 13B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
13.
Its isogeny class
25.1-a
consists of curves linked by isogenies of
degree 13.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.