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([0,1]),K([1,0]),K([40957,-65532]),K([6121178,-3308745])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([1,0]),Polrev([40957,-65532]),Polrev([6121178,-3308745])], K);
magma: E := EllipticCurve([K![1,0],K![0,1],K![1,0],K![40957,-65532],K![6121178,-3308745]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-126a+42)\) | = | \((-2a+1)^{2}\cdot(2)\cdot(-3a+1)^{2}\cdot(3a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 12348 \) | = | \(3^{2}\cdot4\cdot7^{2}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-243855360a+218792448)\) | = | \((-2a+1)^{6}\cdot(2)^{9}\cdot(-3a+1)^{8}\cdot(3a-2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 53981860730241024 \) | = | \(3^{6}\cdot4^{9}\cdot7^{8}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2251439055699625}{25088} \) | ||
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 | $\left(-\frac{20425}{169} a - \frac{5115}{169} : \frac{132681}{2197} a + \frac{38671}{2197} : 1\right)$ |
| Height | \(2.6371586820066923658648112141054261963\) |
| 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: | $\left(-121 a - \frac{121}{4} : \frac{121}{2} a + \frac{117}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 2.6371586820066923658648112141054261963 \) | ||
| Period: | \( 0.095515840159236186195023997760064245896 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot1\cdot2^{2}\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.3268640946961499941607468424048285161 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((2)\) | \(4\) | \(1\) | \(I_{9}\) | Non-split multiplicative | \(1\) | \(1\) | \(9\) | \(9\) |
| \((-3a+1)\) | \(7\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
| \((3a-2)\) | \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
12348.2-a
consists of curves linked by isogenies of
degrees dividing 18.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.