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([-1,3,7,-2,-3,1]),K([0,3,6,-2,-3,1]),K([-5,10,9,-6,-2,1]),K([-34,272,-493,-48,238,-62]),K([353,-2482,3428,611,-1710,422])])
gp: E = ellinit([Polrev([-1,3,7,-2,-3,1]),Polrev([0,3,6,-2,-3,1]),Polrev([-5,10,9,-6,-2,1]),Polrev([-34,272,-493,-48,238,-62]),Polrev([353,-2482,3428,611,-1710,422])], K);
magma: E := EllipticCurve([K![-1,3,7,-2,-3,1],K![0,3,6,-2,-3,1],K![-5,10,9,-6,-2,1],K![-34,272,-493,-48,238,-62],K![353,-2482,3428,611,-1710,422]]);
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: | \((-3a^5+9a^4+9a^3-24a^2-18a)\) | = | \((a^3-a^2-3a)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 19683 \) | = | \(3^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -4619763599211458927 a^{5} + 4368795675516716792 a^{4} + 22834216279555443787 a^{3} + 711261738874157588 a^{2} - 12398128138840295807 a + 2248798976045317301 \) | ||
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);
| |
Torsion generator: | $\left(\frac{4}{3} a^{5} - \frac{14}{3} a^{4} - \frac{1}{3} a^{3} + \frac{31}{3} a^{2} - \frac{8}{3} a + 1 : 2 a^{5} - \frac{29}{3} a^{4} + \frac{22}{3} a^{3} + \frac{52}{3} a^{2} - 21 a + \frac{11}{3} : 1\right)$ |
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: | \( 14894.518564546453600809306043129005704 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 1.39994 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-3a)\) | \(3\) | \(1\) | \(IV^{*}\) | Additive | \(1\) | \(3\) | \(9\) | \(0\) |
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 and 9.
Its isogeny class
27.1-m
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.