Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,1,-3,0,1,0]),K([5,-1,-5,4,1,-1]),K([0,1,0,0,0,0]),K([9,-9,-43,31,11,-8]),K([21,14,-113,63,25,-21])])
gp: E = ellinit([Polrev([-1,1,-3,0,1,0]),Polrev([5,-1,-5,4,1,-1]),Polrev([0,1,0,0,0,0]),Polrev([9,-9,-43,31,11,-8]),Polrev([21,14,-113,63,25,-21])], K);
magma: E := EllipticCurve([K![-1,1,-3,0,1,0],K![5,-1,-5,4,1,-1],K![0,1,0,0,0,0],K![9,-9,-43,31,11,-8],K![21,14,-113,63,25,-21]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-4a^2+a+2)\) | = | \((a^4-4a^2+a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 7 \) | = | \(7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-5a^5-3a^4+37a^3+4a^2-69a+9)\) | = | \((a^4-4a^2+a+2)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -282475249 \) | = | \(-7^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{6378555847697446312}{282475249} a^{5} - \frac{2474411833285999287}{282475249} a^{4} + \frac{28471855515534194816}{282475249} a^{3} + \frac{13996124332632398039}{282475249} a^{2} - \frac{12521297730192365783}{282475249} a - \frac{4603543349670581566}{282475249} \) | ||
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: | $\left(\frac{1}{4} a^{5} + \frac{3}{2} a^{4} - \frac{7}{4} a^{3} - \frac{11}{2} a^{2} + \frac{13}{4} a + \frac{3}{4} : -\frac{5}{4} a^{5} + \frac{15}{8} a^{4} + \frac{25}{8} a^{3} - \frac{67}{8} a^{2} + 2 a + \frac{9}{8} : 1\right)$ |
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: | \( 1353.9010523918115285750093386391544493 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 0.879334 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4-4a^2+a+2)\) | \(7\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
7.1-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.