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([0,1]),K([0,1]),K([1,1]),K([18103,154]),K([-248987,-4963])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,1]),Polrev([1,1]),Polrev([18103,154]),Polrev([-248987,-4963])], K);
magma: E := EllipticCurve([K![0,1],K![0,1],K![1,1],K![18103,154],K![-248987,-4963]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-258a+60)\) | = | \((-2a+1)^{2}\cdot(2)\cdot(3a-2)^{2}\cdot(6a-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 54684 \) | = | \(3^{2}\cdot4\cdot7^{2}\cdot31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-12568715329536a-405984937771008)\) | = | \((-2a+1)^{8}\cdot(2)^{20}\cdot(3a-2)^{7}\cdot(6a-5)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 170084451412887338716546203648 \) | = | \(3^{8}\cdot4^{20}\cdot7^{7}\cdot31^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{1677804026195491}{210138884472832} a + \frac{1018430192992530617}{630416653418496} \) | ||
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);
| |
Torsion generator: | $\left(-157 a + 72 : 42 a - 79 : 1\right)$ |
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: | \( 0.050435338469995673618514763627339620045 \) | ||
Tamagawa product: | \( 160 \) = \(2^{2}\cdot( 2^{2} \cdot 5 )\cdot2\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.3295084993857512166509051138818567990 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a+1)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
\((2)\) | \(4\) | \(20\) | \(I_{20}\) | Split multiplicative | \(-1\) | \(1\) | \(20\) | \(20\) |
\((3a-2)\) | \(7\) | \(2\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
\((6a-5)\) | \(31\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
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 |
\(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
54684.6-e
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.