Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-9,4,14,1,-2]),K([-1,-10,8,21,1,-3]),K([-6,-29,17,43,2,-6]),K([-7,-24,24,43,1,-6]),K([-14,-42,37,72,2,-10])])
gp: E = ellinit([Polrev([-2,-9,4,14,1,-2]),Polrev([-1,-10,8,21,1,-3]),Polrev([-6,-29,17,43,2,-6]),Polrev([-7,-24,24,43,1,-6]),Polrev([-14,-42,37,72,2,-10])], K);
magma: E := EllipticCurve([K![-2,-9,4,14,1,-2],K![-1,-10,8,21,1,-3],K![-6,-29,17,43,2,-6],K![-7,-24,24,43,1,-6],K![-14,-42,37,72,2,-10]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 64 \) | = | \(64\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2)\) | = | \((2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 64 \) | = | \(64\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{189}{2} \) | ||
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: | \(\Z/13\Z\) |
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: | $\left(7 a^{5} - 2 a^{4} - 50 a^{3} - 22 a^{2} + 30 a + 8 : 6 a^{5} - 2 a^{4} - 43 a^{3} - 17 a^{2} + 29 a + 7 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 123898.92720391256997198289982472417215 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(13\) | ||
Leading coefficient: | \( 1.33823 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(64\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(13\) | 13B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
13.
Its isogeny class
64.1-a
consists of curves linked by isogenies of
degree 13.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:
Base field | Curve |
---|---|
\(\Q(\sqrt{5}) \) | a curve with conductor norm 9604 (not in the database) |