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,-10,4,14,1,-2]),K([10,32,-22,-50,-2,7]),K([-5,-18,13,29,1,-4]),K([16,51,-34,-79,-3,11]),K([8,47,-37,-55,1,7])])
gp: E = ellinit([Polrev([-2,-10,4,14,1,-2]),Polrev([10,32,-22,-50,-2,7]),Polrev([-5,-18,13,29,1,-4]),Polrev([16,51,-34,-79,-3,11]),Polrev([8,47,-37,-55,1,7])], K);
magma: E := EllipticCurve([K![-2,-10,4,14,1,-2],K![10,32,-22,-50,-2,7],K![-5,-18,13,29,1,-4],K![16,51,-34,-79,-3,11],K![8,47,-37,-55,1,7]]);
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: | \((-256)\) | = | \((2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 281474976710656 \) | = | \(64^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1323}{256} \) | ||
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: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(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: | \( 314.52135060926873611104817779637933617 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.14823 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(64\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 64.1-b consists of this curve only.
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) |