Base field 4.4.4205.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -5, -1, 1]))
gp: K = nfinit(Polrev([1, -1, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-2,1,0]),K([-2,-1,1,0]),K([0,3,2,-1]),K([-12,35,14,-9]),K([-22,55,23,-14])])
gp: E = ellinit([Polrev([-2,-2,1,0]),Polrev([-2,-1,1,0]),Polrev([0,3,2,-1]),Polrev([-12,35,14,-9]),Polrev([-22,55,23,-14])], K);
magma: E := EllipticCurve([K![-2,-2,1,0],K![-2,-1,1,0],K![0,3,2,-1],K![-12,35,14,-9],K![-22,55,23,-14]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^3+3a^2+8a)\) | = | \((-2a^3+3a^2+8a)\) |
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: | \((-a^3+a^2+6a+12)\) | = | \((-2a^3+3a^2+8a)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 15625 \) | = | \(25^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{15765666}{125} a^{3} + \frac{39249406}{125} a^{2} + \frac{15544741}{125} a + \frac{1058539}{125} \) | ||
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 | $\left(145 a^{3} - 239 a^{2} - 570 a + 224 : 3102 a^{3} - 5109 a^{2} - 12205 a + 4794 : 1\right)$ |
| Height | \(0.48337152294822429867320990191133879314\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{3} + a^{2} + 5 a : -a^{3} + a^{2} + 5 a + 2 : 1\right)$ |
| 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: | \( 0.48337152294822429867320990191133879314 \) | ||
| Period: | \( 765.00307766028140437955179583005706774 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 1.90081560465370 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^3+3a^2+8a)\) | \(25\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
25.2-c
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.