Base field 5.5.24217.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} - x^{2} + 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([4,3,-9,-1,2]),K([-6,-2,14,1,-3]),K([3,0,-9,0,2]),K([-1,0,4,0,-1]),K([0,-1,-1,0,0])])
gp: E = ellinit([Polrev([4,3,-9,-1,2]),Polrev([-6,-2,14,1,-3]),Polrev([3,0,-9,0,2]),Polrev([-1,0,4,0,-1]),Polrev([0,-1,-1,0,0])], K);
magma: E := EllipticCurve([K![4,3,-9,-1,2],K![-6,-2,14,1,-3],K![3,0,-9,0,2],K![-1,0,4,0,-1],K![0,-1,-1,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^4+a^3+10a^2-2a-4)\) | = | \((-2a^4+a^3+10a^2-2a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-28a^4-a^3+51a^2-22)\) | = | \((-2a^4+a^3+10a^2-2a-4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -29 \) | = | \(-29\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{8427382}{29} a^{4} + \frac{7718790}{29} a^{3} - 1223286 a^{2} - \frac{41118125}{29} a - \frac{10576765}{29} \) | ||
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: | \(\Z/7\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(2 a^{4} - a^{3} - 9 a^{2} + 2 a + 4 : -a^{4} + 4 a^{2} + a - 2 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 7344.0836017144368089321061593235450130 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(7\) | ||
| Leading coefficient: | \( 0.963122128 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^4+a^3+10a^2-2a-4)\) | \(29\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
29.1-b
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.