Base field \(\Q(\zeta_{9})^+\)
Generator \(a\), with minimal polynomial \( x^{3} - 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 0, 1]))
gp: K = nfinit(Polrev([-1, -3, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,0,1]),K([1,-1,-1]),K([-2,1,1]),K([-34,-4,10]),K([15,0,-5])])
gp: E = ellinit([Polrev([-1,0,1]),Polrev([1,-1,-1]),Polrev([-2,1,1]),Polrev([-34,-4,10]),Polrev([15,0,-5])], K);
magma: E := EllipticCurve([K![-1,0,1],K![1,-1,-1],K![-2,1,1],K![-34,-4,10],K![15,0,-5]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^2-2)\) | = | \((-a^2+1)\cdot(2a^2-a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 51 \) | = | \(3\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-36a^2-63a+99)\) | = | \((-a^2+1)^{8}\cdot(2a^2-a-3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1896129 \) | = | \(3^{8}\cdot17^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{41087226515522}{7803} a^{2} - \frac{20983089393023}{2601} a - \frac{26817760341455}{7803} \) | ||
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: | \(\Z/2\Z\oplus\Z/4\Z\) | |
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 generators: | $\left(\frac{11}{4} a^{2} - \frac{5}{4} a - \frac{29}{4} : \frac{3}{8} a^{2} - \frac{5}{8} a - 2 : 1\right)$ | $\left(2 a^{2} + a - 2 : -7 a - 11 : 1\right)$ |
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: | \( 94.492928310209949976826709176270048405 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 0.65620089104312465261685214705743089170 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^2+1)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
\((2a^2-a-3)\) | \(17\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
51.2-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.