Base field 6.6.1397493.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, 3, 10, -3, -3, 1]))
gp: K = nfinit(Polrev([1, -6, 3, 10, -3, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 3, 10, -3, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-2,-2,1,0,0]),K([-2,10,7,-6,-2,1]),K([-1,0,1,0,0,0]),K([-9,14,35,-11,-12,4]),K([7,-1,-10,2,3,-1])])
gp: E = ellinit([Polrev([2,-2,-2,1,0,0]),Polrev([-2,10,7,-6,-2,1]),Polrev([-1,0,1,0,0,0]),Polrev([-9,14,35,-11,-12,4]),Polrev([7,-1,-10,2,3,-1])], K);
magma: E := EllipticCurve([K![2,-2,-2,1,0,0],K![-2,10,7,-6,-2,1],K![-1,0,1,0,0,0],K![-9,14,35,-11,-12,4],K![7,-1,-10,2,3,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^5+3a^4+3a^3-8a^2-6a)\) | = | \((a^3-a^2-3a)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-a^3+3a+1)\) | = | \((a^3-a^2-3a)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 27 \) | = | \(3^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -607008006 a^{5} + 193715388 a^{4} + 2340294876 a^{3} + 203925438 a^{2} - 1274124573 a + 226434609 \) | ||
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: | \(\Z/3\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generator: | $\left(-\frac{1}{3} a^{4} + \frac{4}{3} a^{3} - \frac{8}{3} a + \frac{2}{3} : \frac{1}{3} a^{4} - 2 a^{3} + a^{2} + \frac{14}{3} a - 3 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(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: | \( 17180.226605878382826046126270597151993 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 1.61477 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-3a)\) | \(3\) | \(1\) | \(II\) | Additive | \(1\) | \(3\) | \(3\) | \(0\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
27.1-h
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.