Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([0,1]),K([-410,97]),K([2978,-980])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([-410,97]),Polrev([2978,-980])], K);
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![0,1],K![-410,97],K![2978,-980]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-285a+114)\) | = | \((-2a+1)^{2}\cdot(-5a+3)\cdot(-5a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 61731 \) | = | \(3^{2}\cdot19\cdot19^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-442123920a+685204353)\) | = | \((-2a+1)^{10}\cdot(-5a+3)^{2}\cdot(-5a+2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 362033331456891249 \) | = | \(3^{10}\cdot19^{2}\cdot19^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{30664297}{3249} \) | ||
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: | $\left(3 a - 23 : 8 a + 13 : 1\right)$ | $\left(-a + 9 : -4 a - 5 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 0.43241862138931701517305741462916006311 \) | ||
Tamagawa product: | \( 32 \) = \(2^{2}\cdot2\cdot2^{2}\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 0.99862802984691617642153269630275646570 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a+1)\) | \(3\) | \(4\) | \(I_{4}^{*}\) | Additive | \(-1\) | \(2\) | \(10\) | \(4\) |
\((-5a+3)\) | \(19\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((-5a+2)\) | \(19\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
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
61731.3-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.