Base field 6.6.1312625.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 7 x^{3} + 12 x^{2} - 12 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -12, 12, 7, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -12, 12, 7, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -12, 12, 7, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,9,0,-6,0,1]),K([2,-8,-4,6,1,-1]),K([-3,9,1,-6,0,1]),K([245,3138,-911,-2302,192,366]),K([3127,39749,-10909,-29436,2285,4708])])
gp: E = ellinit([Polrev([0,9,0,-6,0,1]),Polrev([2,-8,-4,6,1,-1]),Polrev([-3,9,1,-6,0,1]),Polrev([245,3138,-911,-2302,192,366]),Polrev([3127,39749,-10909,-29436,2285,4708])], K);
magma: E := EllipticCurve([K![0,9,0,-6,0,1],K![2,-8,-4,6,1,-1],K![-3,9,1,-6,0,1],K![245,3138,-911,-2302,192,366],K![3127,39749,-10909,-29436,2285,4708]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^5+6a^3+a^2-8a-2)\) | = | \((-a^5+6a^3+a^2-8a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 4 \) | = | \(4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^5+a^4-9a^3-6a^2+16a+4)\) | = | \((-a^5+6a^3+a^2-8a-2)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 4096 \) | = | \(4^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{10576507188547681123}{64} a^{5} - \frac{2616146338735194005}{32} a^{4} + \frac{16553698917074597961}{16} a^{3} + \frac{24936296834825976631}{64} a^{2} - \frac{89645580404194791495}{64} a - \frac{3537982172908461591}{32} \) | ||
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: | \(\Z/2\Z\) |
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: | $\left(\frac{33}{4} a^{5} + \frac{5}{2} a^{4} - 51 a^{3} - \frac{49}{4} a^{2} + 68 a + \frac{13}{4} : -\frac{41}{8} a^{5} - \frac{17}{8} a^{4} + \frac{263}{8} a^{3} + \frac{19}{2} a^{2} - \frac{191}{4} a - \frac{21}{8} : 1\right)$ |
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: | \( 2799.3372214788879792328427346169253742 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.22167 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^5+6a^3+a^2-8a-2)\) | \(4\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
4.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.