Base field \(\Q(\zeta_{21})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -8, 8, 6, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -8, 8, 6, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 8, 6, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,6,-2,-5,1,1]),K([-1,2,-2,-4,1,1]),K([1,0,-3,0,1,0]),K([-11,60,-20,-72,11,17]),K([-13,133,-84,-144,34,34])])
gp: E = ellinit([Polrev([-1,6,-2,-5,1,1]),Polrev([-1,2,-2,-4,1,1]),Polrev([1,0,-3,0,1,0]),Polrev([-11,60,-20,-72,11,17]),Polrev([-13,133,-84,-144,34,34])], K);
magma: E := EllipticCurve([K![-1,6,-2,-5,1,1],K![-1,2,-2,-4,1,1],K![1,0,-3,0,1,0],K![-11,60,-20,-72,11,17],K![-13,133,-84,-144,34,34]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4+2a^3-4a^2-6a+3)\) | = | \((a^4+2a^3-4a^2-6a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2a^5+10a^3+3a^2-9a-8)\) | = | \((a^4+2a^3-4a^2-6a+3)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -68921 \) | = | \(-41^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1824869065509}{68921} a^{5} + \frac{491296262031}{68921} a^{4} + \frac{11308085774361}{68921} a^{3} - \frac{2685847365072}{68921} a^{2} - \frac{16561220551596}{68921} a + \frac{2497339386342}{68921} \) | ||
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 | $\left(-a^{5} + 4 a^{3} - 5 a + 3 : 2 a^{5} + a^{4} - 11 a^{3} + 3 a^{2} + 8 a - 4 : 1\right)$ |
Height | \(0.0013920951906400464923876960932390682045\) |
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: | \( 0.0013920951906400464923876960932390682045 \) | ||
Period: | \( 59892.466146077458345329072565085321461 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.22786 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4+2a^3-4a^2-6a+3)\) | \(41\) | \(3\) | \(I_{3}\) | 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
41.5-c
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.