Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-1,-3,0,1,0]),K([1,1,2,-4,-1,1]),K([-2,8,7,-9,-2,2]),K([-5,1,16,0,-4,0]),K([-12,9,18,-17,-5,4])])
gp: E = ellinit([Polrev([1,-1,-3,0,1,0]),Polrev([1,1,2,-4,-1,1]),Polrev([-2,8,7,-9,-2,2]),Polrev([-5,1,16,0,-4,0]),Polrev([-12,9,18,-17,-5,4])], K);
magma: E := EllipticCurve([K![1,-1,-3,0,1,0],K![1,1,2,-4,-1,1],K![-2,8,7,-9,-2,2],K![-5,1,16,0,-4,0],K![-12,9,18,-17,-5,4]]);
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: | \((-8)\) | = | \((2)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -262144 \) | = | \(-64^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{372514795}{2} a^{5} + \frac{1962808611}{8} a^{4} + \frac{3650149817}{4} a^{3} - \frac{6936306195}{8} a^{2} - \frac{7715765979}{8} a + \frac{2182489307}{8} \) | ||
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: | \(1\) |
Generator | $\left(-5 a^{5} + 8 a^{4} + 22 a^{3} - 31 a^{2} - 17 a + 18 : 16 a^{5} - 29 a^{4} - 71 a^{3} + 115 a^{2} + 59 a - 69 : 1\right)$ |
Height | \(0.0077837535249815245985547087823022324145\) |
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: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.0077837535249815245985547087823022324145 \) | ||
Period: | \( 13990.337260106950258133421748028974557 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.81425 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(64\) | \(3\) | \(I_{3}\) | 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 |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
64.1-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.