Base field 5.5.36497.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 5, -3, -2, 1]))
gp: K = nfinit(Polrev([-1, 1, 5, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,6,-3,-2,1]),K([2,5,0,-1,0]),K([-2,0,1,0,0]),K([-21,121,-16,-48,7]),K([131,-300,-11,130,-2])])
gp: E = ellinit([Polrev([2,6,-3,-2,1]),Polrev([2,5,0,-1,0]),Polrev([-2,0,1,0,0]),Polrev([-21,121,-16,-48,7]),Polrev([131,-300,-11,130,-2])], K);
magma: E := EllipticCurve([K![2,6,-3,-2,1],K![2,5,0,-1,0],K![-2,0,1,0,0],K![-21,121,-16,-48,7],K![131,-300,-11,130,-2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-2a^3-2a^2+4a+1)\) | = | \((a^4-2a^3-2a^2+4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 37 \) | = | \(37\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^4-6a^3-9a^2+18a+3)\) | = | \((a^4-2a^3-2a^2+4a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -50653 \) | = | \(-37^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1272945793794372}{50653} a^{4} + \frac{859591705080800}{50653} a^{3} + \frac{4922533480405137}{50653} a^{2} + \frac{233798144853843}{50653} a - \frac{948239678172729}{50653} \) | ||
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 | $\left(a^{4} - 2 a^{3} + 3 a - 4 : 6 a^{4} - 16 a^{3} - 13 a^{2} + 38 a - 6 : 1\right)$ |
| Height | \(0.0098697986302844935643455617563804060516\) |
| 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: | \( 0.0098697986302844935643455617563804060516 \) | ||
| Period: | \( 6342.5736039157764152450411213116277979 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.63838223 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-2a^3-2a^2+4a+1)\) | \(37\) | \(1\) | \(I_{3}\) | Non-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 |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
37.1-d
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.