Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-5,0,1]),K([-1,-4,0,1]),K([-2,-1,1,0]),K([-6,-20,-2,5]),K([12,7,-1,-1])])
gp: E = ellinit([Polrev([-1,-5,0,1]),Polrev([-1,-4,0,1]),Polrev([-2,-1,1,0]),Polrev([-6,-20,-2,5]),Polrev([12,7,-1,-1])], K);
magma: E := EllipticCurve([K![-1,-5,0,1],K![-1,-4,0,1],K![-2,-1,1,0],K![-6,-20,-2,5],K![12,7,-1,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-2a^2-4a+5)\) | = | \((-a^3+2a^2+3a-4)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^3-3a^2-7a-2)\) | = | \((-a^3+2a^2+3a-4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 625 \) | = | \(5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 4036500 a^{3} - 7580142 a^{2} - 9865638 a + 6254145 \) | ||
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 | $\left(3 a^{2} + 5 a : -8 a^{3} - 22 a^{2} - 2 a + 18 : 1\right)$ |
| Height | \(0.070938334656423382592893651979731163225\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{2}{3} a^{3} + \frac{2}{3} a^{2} + 3 a : \frac{1}{3} a^{3} - \frac{4}{3} a^{2} + \frac{1}{3} a + 5 : 1\right)$ |
| 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: | \( 0.070938334656423382592893651979731163225 \) | ||
| Period: | \( 2779.6857804987638503309888109580588007 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 2.64119530524228 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+2a^2+3a-4)\) | \(5\) | \(3\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
25.1-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.