Base field 5.5.70601.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0,0]),K([0,0,4,1,-1]),K([2,3,-5,-1,1]),K([-25,5,39,3,-7]),K([25,8,-46,-7,9])])
gp: E = ellinit([Polrev([1,1,0,0,0]),Polrev([0,0,4,1,-1]),Polrev([2,3,-5,-1,1]),Polrev([-25,5,39,3,-7]),Polrev([25,8,-46,-7,9])], K);
magma: E := EllipticCurve([K![1,1,0,0,0],K![0,0,4,1,-1],K![2,3,-5,-1,1],K![-25,5,39,3,-7],K![25,8,-46,-7,9]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^4-2a^3-9a^2+2a+2)\) | = | \((2a^4-2a^3-9a^2+2a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 53 \) | = | \(53\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^4-2a^3-9a^2+2a+2)\) | = | \((2a^4-2a^3-9a^2+2a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -53 \) | = | \(-53\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{9071767}{53} a^{4} + \frac{11769744}{53} a^{3} + \frac{57578323}{53} a^{2} - \frac{6421884}{53} a - \frac{35004004}{53} \) | ||
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 | $\left(5 a^{4} - 4 a^{3} - 25 a^{2} + 3 a + 13 : 10 a^{4} - 6 a^{3} - 53 a^{2} + 34 : 1\right)$ |
| Height | \(0.065935189763161249530014850016313543137\) |
| 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: | \( 0.065935189763161249530014850016313543137 \) | ||
| Period: | \( 2675.8687864083911984238249184849370080 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.32006550 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^4-2a^3-9a^2+2a+2)\) | \(53\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 53.2-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.