Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0,0,0]),K([-2,10,9,-11,-2,2]),K([-2,0,1,0,0,0]),K([1,0,3,0,-1,0]),K([4,-41,-20,39,5,-7])])
gp: E = ellinit([Polrev([1,1,0,0,0,0]),Polrev([-2,10,9,-11,-2,2]),Polrev([-2,0,1,0,0,0]),Polrev([1,0,3,0,-1,0]),Polrev([4,-41,-20,39,5,-7])], K);
magma: E := EllipticCurve([K![1,1,0,0,0,0],K![-2,10,9,-11,-2,2],K![-2,0,1,0,0,0],K![1,0,3,0,-1,0],K![4,-41,-20,39,5,-7]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^5+a^4+12a^3-4a^2-12a-1)\) | = | \((-2a^5+a^4+12a^3-4a^2-12a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((3a^5-a^4-17a^3+5a^2+17a+8)\) | = | \((-2a^5+a^4+12a^3-4a^2-12a-1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -357911 \) | = | \(-71^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{29169505440}{357911} a^{5} + \frac{151117918400}{357911} a^{4} - \frac{25425543508}{357911} a^{3} - \frac{654677096029}{357911} a^{2} + \frac{727216594590}{357911} a - \frac{74279894665}{357911} \) | ||
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);
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Torsion generator: | $\left(-3 a^{5} + 2 a^{4} + 17 a^{3} - 9 a^{2} - 17 a + 2 : 6 a^{5} - 3 a^{4} - 35 a^{3} + 15 a^{2} + 35 a - 3 : 1\right)$ |
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: | \( 55543.943665711012889444769701669017801 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(6\) | ||
Leading coefficient: | \( 1.83952 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a^5+a^4+12a^3-4a^2-12a-1)\) | \(71\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
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
71.1-c
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.