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([0,0]),K([4081,6805]),K([53553,-21740])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,0]),Polrev([4081,6805]),Polrev([53553,-21740])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,0],K![4081,6805],K![53553,-21740]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((90a-240)\) | = | \((-2a+1)^{2}\cdot(2)\cdot(-3a+1)^{2}\cdot(5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 44100 \) | = | \(3^{2}\cdot4\cdot7^{2}\cdot25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-56953125000000a+51099609375000)\) | = | \((-2a+1)^{8}\cdot(2)^{3}\cdot(-3a+1)^{6}\cdot(5)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 2944546085357666015625000000 \) | = | \(3^{8}\cdot4^{3}\cdot7^{6}\cdot25^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{10316097499609}{5859375000} \) | ||
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: | $\left(-\frac{439}{4} a + 87 : \frac{265}{4} a - \frac{787}{8} : 1\right)$ |
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: | \( 0.070609315944893458621398142900677798491 \) | ||
Tamagawa product: | \( 96 \) = \(2^{2}\cdot1\cdot2\cdot( 2^{2} \cdot 3 )\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.9567827632678195053883058266357616528 \) | ||
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 | \(-1\) | \(2\) | \(8\) | \(2\) |
\((2)\) | \(4\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
\((-3a+1)\) | \(7\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
\((5)\) | \(25\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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 |
\(3\) | 3B[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
44100.1-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.