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([-1,2,7,-6,-4,2]),K([-3,1,11,-5,-5,2]),K([2,4,0,-5,-1,1]),K([1,-11,-3,7,1,-1]),K([-4,0,12,-5,-5,2])])
gp: E = ellinit([Polrev([-1,2,7,-6,-4,2]),Polrev([-3,1,11,-5,-5,2]),Polrev([2,4,0,-5,-1,1]),Polrev([1,-11,-3,7,1,-1]),Polrev([-4,0,12,-5,-5,2])], K);
magma: E := EllipticCurve([K![-1,2,7,-6,-4,2],K![-3,1,11,-5,-5,2],K![2,4,0,-5,-1,1],K![1,-11,-3,7,1,-1],K![-4,0,12,-5,-5,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^5+4a^4-11a^2+3a+4)\) | = | \((-a^5+4a^4-11a^2+3a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((7)\) | = | \((-a^5+4a^4-11a^2+3a+4)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 117649 \) | = | \(49^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{2333735}{7} a^{5} - \frac{1536755}{7} a^{4} - 1654035 a^{3} - \frac{3465769}{7} a^{2} + \frac{4986743}{7} a + \frac{1824569}{7} \) | ||
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 | $\left(8 a^{5} - 22 a^{4} - 15 a^{3} + 51 a^{2} - 8 a - 10 : -53 a^{5} + 146 a^{4} + 101 a^{3} - 341 a^{2} + 47 a + 70 : 1\right)$ |
Height | \(0.0018047521437735257265851585826832639131\) |
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: | \( 0.0018047521437735257265851585826832639131 \) | ||
Period: | \( 47699.109079200081911902247417438000449 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.35052 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^5+4a^4-11a^2+3a+4)\) | \(49\) | \(3\) | \(I_{3}\) | 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
49.1-b
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.