Base field \(\Q(\zeta_{15})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -4, -1, 1]))
gp: K = nfinit(Polrev([1, 4, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-2,0,1]),K([2,-3,-1,1]),K([-1,-3,1,1]),K([-18,-97,4,25]),K([-64,-308,20,82])])
gp: E = ellinit([Polrev([0,-2,0,1]),Polrev([2,-3,-1,1]),Polrev([-1,-3,1,1]),Polrev([-18,-97,4,25]),Polrev([-64,-308,20,82])], K);
magma: E := EllipticCurve([K![0,-2,0,1],K![2,-3,-1,1],K![-1,-3,1,1],K![-18,-97,4,25],K![-64,-308,20,82]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^3-2a^2+6a+3)\) | = | \((-2a^3-2a^2+6a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-25a^3+101a-11)\) | = | \((-2a^3-2a^2+6a+3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 923521 \) | = | \(31^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{17801814367097}{923521} a^{3} + \frac{14868316428812}{923521} a^{2} - \frac{44271766433252}{923521} a - \frac{9739909921969}{923521} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(a^{3} - 4 a - 2 : a^{3} - 2 a : 1\right)$ | $\left(\frac{3}{4} a^{3} + \frac{1}{4} a^{2} - \frac{15}{4} a - \frac{7}{4} : \frac{3}{4} a^{3} + \frac{5}{8} a^{2} - \frac{9}{4} a - \frac{1}{8} : 1\right)$ |
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: | \( 226.24024852703479660383539055627057485 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 0.843147624921494 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a^3-2a^2+6a+3)\) | \(31\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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
31.3-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.