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([0,-2,-3,1,1]),K([-5,3,5,-1,-1]),K([3,-2,-4,1,1]),K([-4049,587,7125,-1331,-2681]),K([-132130,126719,354020,-108973,-146736])])
gp: E = ellinit([Polrev([0,-2,-3,1,1]),Polrev([-5,3,5,-1,-1]),Polrev([3,-2,-4,1,1]),Polrev([-4049,587,7125,-1331,-2681]),Polrev([-132130,126719,354020,-108973,-146736])], K);
magma: E := EllipticCurve([K![0,-2,-3,1,1],K![-5,3,5,-1,-1],K![3,-2,-4,1,1],K![-4049,587,7125,-1331,-2681],K![-132130,126719,354020,-108973,-146736]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^4+2a^2+1)\) | = | \((-a^4+2a^2+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 23 \) | = | \(23\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((2a^4-9a^2+2a+4)\) | = | \((-a^4+2a^2+1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -529 \) | = | \(-23^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1665599353208772437646840723861648544}{529} a^{4} - \frac{515871876593405186030634717346598316}{529} a^{3} + \frac{5986748941379712064585808940738486553}{529} a^{2} + \frac{2844175551769835948036386085428144797}{529} a - \frac{1271720281023855543211042033983127776}{529} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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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);
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Torsion generator: | $\left(-\frac{27}{2} a^{4} + \frac{9}{2} a^{3} + \frac{79}{2} a^{2} - \frac{115}{4} a - \frac{109}{4} : \frac{239}{8} a^{4} + 29 a^{3} - \frac{281}{4} a^{2} - \frac{449}{8} a + \frac{57}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 0.31780034143956915037034160225321733000 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+2a^2+1)\) | \(23\) | \(2\) | \(I_{2}\) | 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
23.4-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.