Base field \(\Q(\zeta_{11})^+\)
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -4, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 3, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,1,-4,0,1]),K([-1,-4,1,1,0]),K([3,1,-4,0,1]),K([-761,-1659,-356,731,-32]),K([-8289,-28326,-10713,13981,-179])])
gp: E = ellinit([Polrev([3,1,-4,0,1]),Polrev([-1,-4,1,1,0]),Polrev([3,1,-4,0,1]),Polrev([-761,-1659,-356,731,-32]),Polrev([-8289,-28326,-10713,13981,-179])], K);
magma: E := EllipticCurve([K![3,1,-4,0,1],K![-1,-4,1,1,0],K![3,1,-4,0,1],K![-761,-1659,-356,731,-32],K![-8289,-28326,-10713,13981,-179]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+a^3+3a^2-3a-2)\) | = | \((-a^4+a^3+3a^2-3a-2)\) |
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: | \((-a^4-2a^3+5a^2+4a-4)\) | = | \((-a^4+a^3+3a^2-3a-2)^{2}\) |
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: | \( \frac{369039431219152047702358790473298695}{529} a^{4} - \frac{989950881753394751645286769153540921}{529} a^{3} + \frac{189441628332164246837405561968453764}{529} a^{2} + \frac{788381415458006631258384866252375903}{529} a - \frac{219339008287534382162591638591097088}{529} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(19 a^{4} - \frac{3}{4} a^{3} - \frac{311}{4} a^{2} + \frac{29}{4} a + \frac{179}{4} : -\frac{49}{2} a^{4} + \frac{93}{8} a^{3} + \frac{339}{4} a^{2} - \frac{203}{8} a - \frac{123}{2} : 1\right)$ |
| 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: | \( 0.31780034143956915037034160225321732999 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.820765345 \) | ||
| Analytic order of Ш: | \( 625 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+3a^2-3a-2)\) | \(23\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(2\) | 2B |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
23.3-a
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.