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([1,0]),K([1,0]),K([0,0]),K([7,-6]),K([1,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([0,0]),Polrev([7,-6]),Polrev([1,0])], K);
magma: E := EllipticCurve([K![1,0],K![1,0],K![0,0],K![7,-6],K![1,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((364a-210)\) | = | \((2)\cdot(-3a+1)\cdot(3a-2)^{2}\cdot(9a-8)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 100156 \) | = | \(4\cdot7\cdot7^{2}\cdot73\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4704a+14896)\) | = | \((2)^{4}\cdot(-3a+1)^{3}\cdot(3a-2)^{2}\cdot(9a-8)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 314089216 \) | = | \(4^{4}\cdot7^{3}\cdot7^{2}\cdot73\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{11544123}{200312} a + \frac{622547377}{400624} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-2 a : 4 a - 1 : 1\right)$ |
| Height | \(0.050112547297506526668846124791429251470\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.050112547297506526668846124791429251470 \) | ||
| Period: | \( 2.6961437907563351262387185668780290677 \) | ||
| Tamagawa product: | \( 12 \) = \(2^{2}\cdot3\cdot1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.7442957024974924039585200541743227490 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((-3a+1)\) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((3a-2)\) | \(7\) | \(1\) | \(II\) | Additive | \(-1\) | \(2\) | \(2\) | \(0\) |
| \((9a-8)\) | \(73\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 100156.6-c consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.