Base field 5.5.138136.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 4 x - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 4, 3, -6, -1, 1]))
gp: K = nfinit(Polrev([-2, 4, 3, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 4, 3, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([6,-1,-12,-1,2]),K([4,3,-6,-1,1]),K([9,-3,-18,-1,3]),K([-84,104,147,-7,-22]),K([-164,-5,639,64,-111])])
gp: E = ellinit([Polrev([6,-1,-12,-1,2]),Polrev([4,3,-6,-1,1]),Polrev([9,-3,-18,-1,3]),Polrev([-84,104,147,-7,-22]),Polrev([-164,-5,639,64,-111])], K);
magma: E := EllipticCurve([K![6,-1,-12,-1,2],K![4,3,-6,-1,1],K![9,-3,-18,-1,3],K![-84,104,147,-7,-22],K![-164,-5,639,64,-111]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2-2a-1)\) | = | \((a^2-2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-30a^4+33a^3+139a^2-107a-25)\) | = | \((a^2-2a-1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 887503681 \) | = | \(31^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1835739307342}{29791} a^{4} - \frac{5598566631613}{29791} a^{3} + \frac{490084842930}{29791} a^{2} + \frac{4428529373282}{29791} a - \frac{1764167405540}{29791} \) | ||
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: | \(\Z/6\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-11 a^{4} + 7 a^{3} + 63 a^{2} - 4 a - 16 : -63 a^{4} + 37 a^{3} + 362 a^{2} - 5 a - 94 : 1\right)$ |
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: | \( 4701.8165690048013836419737499652396575 \) | ||
Tamagawa product: | \( 6 \) | ||
Torsion order: | \(6\) | ||
Leading coefficient: | \( 2.10843877 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-2a-1)\) | \(31\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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 |
\(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
31.1-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.