Base field 4.4.2624.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -3, -2, 1]))
gp: K = nfinit(Polrev([1, 2, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-1,-2,1]),K([1,1,-3,1]),K([1,-1,-2,1]),K([-5,2,6,-2]),K([3,-1,-5,2])])
gp: E = ellinit([Polrev([1,-1,-2,1]),Polrev([1,1,-3,1]),Polrev([1,-1,-2,1]),Polrev([-5,2,6,-2]),Polrev([3,-1,-5,2])], K);
magma: E := EllipticCurve([K![1,-1,-2,1],K![1,1,-3,1],K![1,-1,-2,1],K![-5,2,6,-2],K![3,-1,-5,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-3a+1)\) | = | \((a^3-a^2-3a+1)\) |
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: | \((-38a^3+79a^2+66a-40)\) | = | \((a^3-a^2-3a+1)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 9765625 \) | = | \(25^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{33914163}{3125} a^{3} + \frac{86534812}{3125} a^{2} + \frac{62225714}{3125} a - \frac{23022392}{625} \) | ||
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 | $\left(-a^{3} + a^{2} + 3 a + 2 : -3 a^{3} + 5 a^{2} + 10 a + 2 : 1\right)$ |
| Height | \(0.0032006178572261560013634276145926737861\) |
| 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: | \( 0.0032006178572261560013634276145926737861 \) | ||
| Period: | \( 896.14501299225505902963217688179714023 \) | ||
| Tamagawa product: | \( 5 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.11985088251119 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-3a+1)\) | \(25\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 25.2-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.