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([0,0]),K([0,0]),K([0,-714]),K([-82908,0])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,0]),Polrev([0,-714]),Polrev([-82908,0])], K);
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,0],K![0,-714],K![-82908,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-364a+182)\) | = | \((-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(-4a+1)\cdot(4a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 99372 \) | = | \(3\cdot4\cdot7\cdot7\cdot13\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2997011332224)\) | = | \((-2a+1)^{14}\cdot(2)^{7}\cdot(-3a+1)^{7}\cdot(3a-2)^{7}\cdot(-4a+1)\cdot(4a-3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 8982076925479075300786176 \) | = | \(3^{14}\cdot4^{7}\cdot7^{7}\cdot7^{7}\cdot13\cdot13\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{40251338884511}{2997011332224} \) | ||
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: | \(1\) |
Generator | $\left(-105 a + 105 : -1323 a + 609 : 1\right)$ |
Height | \(0.22894617307898607580336556239510434561\) |
Torsion structure: | \(\Z/7\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(336 a - 336 : -6006 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.22894617307898607580336556239510434561 \) | ||
Period: | \( 0.11681249938886455880748121481189896431 \) | ||
Tamagawa product: | \( 4802 \) = \(( 2 \cdot 7 )\cdot7\cdot7\cdot7\cdot1\cdot1\) | ||
Torsion order: | \(7\) | ||
Leading coefficient: | \( 6.0526860053491289052846462772023922835 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a+1)\) | \(3\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
\((2)\) | \(4\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
\((-3a+1)\) | \(7\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
\((3a-2)\) | \(7\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
\((-4a+1)\) | \(13\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((4a-3)\) | \(13\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(7\) | 7B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
99372.5-j
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 546.f2 |
\(\Q\) | 1638.h2 |