Base field 3.3.169.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -4, -1, 1]))
gp: K = nfinit(Polrev([-1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1]),K([-4,0,1]),K([-3,-1,1]),K([-3394,-1177,929]),K([-78558,-27403,21516])])
gp: E = ellinit([Polrev([-3,0,1]),Polrev([-4,0,1]),Polrev([-3,-1,1]),Polrev([-3394,-1177,929]),Polrev([-78558,-27403,21516])], K);
magma: E := EllipticCurve([K![-3,0,1],K![-4,0,1],K![-3,-1,1],K![-3394,-1177,929],K![-78558,-27403,21516]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-3a-4)\) | = | \((a^2-3a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-17885a^2-28211a+75240)\) | = | \((a^2-3a-4)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -819628286980801 \) | = | \(-31^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1475484619244572890625}{819628286980801} a^{2} + \frac{3509758045748731549120}{819628286980801} a + \frac{1066714472957007302564}{819628286980801} \) | ||
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: | \( 1.4393913935995161893122560470467492203 \) | ||
| Tamagawa product: | \( 10 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.1072241489227047610094277284974994002 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-3a-4)\) | \(31\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
31.3-b
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.