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([-3,8,7,-5,-2,1]),K([-1,10,7,-6,-2,1]),K([0,-2,-1,1,0,0]),K([-7,33,29,-23,-10,5]),K([-3,13,14,-7,-4,2])])
gp: E = ellinit([Polrev([-3,8,7,-5,-2,1]),Polrev([-1,10,7,-6,-2,1]),Polrev([0,-2,-1,1,0,0]),Polrev([-7,33,29,-23,-10,5]),Polrev([-3,13,14,-7,-4,2])], K);
magma: E := EllipticCurve([K![-3,8,7,-5,-2,1],K![-1,10,7,-6,-2,1],K![0,-2,-1,1,0,0],K![-7,33,29,-23,-10,5],K![-3,13,14,-7,-4,2]]);
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: | \((3a^5-9a^4-6a^3+24a^2-3)\) | = | \((a^3-a^2-3a)^{9}\) |
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: | \( 59860377 a^{5} - 144641619 a^{4} - 263914785 a^{3} + 444199410 a^{2} + 438961572 a - 102352338 \) | ||
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 | $\left(a^{5} - 4 a^{4} + 2 a^{3} + 7 a^{2} - 8 a + 1 : -2 a^{5} + 9 a^{4} - 6 a^{3} - 16 a^{2} + 19 a - 3 : 1\right)$ |
Height | \(0.034078774406413841674109693728738132561\) |
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: | $\left(-\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + 2 a^{3} - \frac{2}{3} a^{2} - \frac{7}{3} a : \frac{2}{3} a^{5} - \frac{4}{3} a^{4} - 3 a^{3} + \frac{13}{3} a^{2} + \frac{13}{3} a - 1 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(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: | \( 0.034078774406413841674109693728738132561 \) | ||
Period: | \( 56479.063644706498032255049727204084948 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 3.25632 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3-a^2-3a)\) | \(3\) | \(3\) | \(IV^{*}\) | 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.
Its isogeny class
27.1-p
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.