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([-3,-2,5,1,-1]),K([-8,-2,15,1,-3]),K([0,0,0,0,0]),K([-6,-2,10,2,-2]),K([45,13,-81,-6,17])])
gp: E = ellinit([Polrev([-3,-2,5,1,-1]),Polrev([-8,-2,15,1,-3]),Polrev([0,0,0,0,0]),Polrev([-6,-2,10,2,-2]),Polrev([45,13,-81,-6,17])], K);
magma: E := EllipticCurve([K![-3,-2,5,1,-1],K![-8,-2,15,1,-3],K![0,0,0,0,0],K![-6,-2,10,2,-2],K![45,13,-81,-6,17]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-3a)\) | = | \((a^3-3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 23 \) | = | \(23\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4a^4-2a^3-17a^2+4a+6)\) | = | \((a^3-3a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -529 \) | = | \(-23^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2370155}{529} a^{4} + \frac{9772390}{529} a^{3} - \frac{30902810}{529} a^{2} - \frac{40834965}{529} a - \frac{14158252}{529} \) | ||
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 | $\left(-6 a^{4} + 5 a^{3} + 26 a^{2} - 17 a - 5 : -30 a^{4} + 23 a^{3} + 134 a^{2} - 73 a - 37 : 1\right)$ |
| Height | \(0.00065557609631294353761010235442956723507\) |
| 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: | \( 0.00065557609631294353761010235442956723507 \) | ||
| Period: | \( 29464.428972202316895788878653096550926 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.24125488 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a)\) | \(23\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 23.1-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.