Base field 6.6.434581.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,3,1,-5,-1,1]),K([2,-5,-4,4,2,-1]),K([2,3,-3,-2,1,0]),K([5,3,-2,-3,1,0]),K([5,9,-1,-8,0,1])])
gp: E = ellinit([Polrev([1,3,1,-5,-1,1]),Polrev([2,-5,-4,4,2,-1]),Polrev([2,3,-3,-2,1,0]),Polrev([5,3,-2,-3,1,0]),Polrev([5,9,-1,-8,0,1])], K);
magma: E := EllipticCurve([K![1,3,1,-5,-1,1],K![2,-5,-4,4,2,-1],K![2,3,-3,-2,1,0],K![5,3,-2,-3,1,0],K![5,9,-1,-8,0,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((3a^5-7a^4-9a^3+17a^2+3a-5)\) | = | \((3a^5-7a^4-9a^3+17a^2+3a-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^5-3a^4-2a^3+9a^2-a-3)\) | = | \((3a^5-7a^4-9a^3+17a^2+3a-5)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -29 \) | = | \(-29\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{48945886}{29} a^{5} + \frac{138712716}{29} a^{4} - \frac{41434329}{29} a^{3} - \frac{133295612}{29} a^{2} + \frac{35939234}{29} a + \frac{27409508}{29} \) | ||
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 | $\left(3 a^{4} - 5 a^{3} - 13 a^{2} + 10 a + 11 : -3 a^{5} - 5 a^{4} + 30 a^{3} + 33 a^{2} - 46 a - 39 : 1\right)$ |
Height | \(0.0023295865454728247176156982656608317891\) |
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: | \( 0.0023295865454728247176156982656608317891 \) | ||
Period: | \( 95834.381593173948413872543324347624021 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.03196 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3a^5-7a^4-9a^3+17a^2+3a-5)\) | \(29\) | \(1\) | \(I_{1}\) | 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 |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
29.2-d
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.