Base field 4.4.2225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 2 x + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([4, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-5/2,1/2,1/2]),K([0,-5/2,-1/2,1/2]),K([-2,-5/2,1/2,1/2]),K([-17,-25,0,5]),K([-45,-62,2,13])])
gp: E = ellinit([Polrev([-3,-5/2,1/2,1/2]),Polrev([0,-5/2,-1/2,1/2]),Polrev([-2,-5/2,1/2,1/2]),Polrev([-17,-25,0,5]),Polrev([-45,-62,2,13])], K);
magma: E := EllipticCurve([K![-3,-5/2,1/2,1/2],K![0,-5/2,-1/2,1/2],K![-2,-5/2,1/2,1/2],K![-17,-25,0,5],K![-45,-62,2,13]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-5)\) | = | \((a^2-5)\) |
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: | \((-2a^3+4a^2+6a-5)\) | = | \((a^2-5)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 841 \) | = | \(29^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{176483079}{841} a^{3} + \frac{478215419}{841} a^{2} + \frac{40825114}{841} a - \frac{381703735}{841} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3 : a^{3} - 5 a - 4 : 1\right)$ | $\left(\frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{17}{8} a - \frac{3}{2} : \frac{7}{8} a^{3} - 4 a - \frac{23}{8} : 1\right)$ |
| 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: | \( 577.80066295441714626101939027203390666 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.53116869449488 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-5)\) | \(29\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
29.2-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.