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([0,-1]),K([0,0]),K([97,-97]),K([1337,0])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([97,-97]),Polrev([1337,0])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![97,-97],K![1337,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-308a+154)\) | = | \((-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(11)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 71148 \) | = | \(3\cdot4\cdot7\cdot7\cdot121\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-723148272)\) | = | \((-2a+1)^{4}\cdot(2)^{4}\cdot(-3a+1)^{3}\cdot(3a-2)^{3}\cdot(11)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 522943423296585984 \) | = | \(3^{4}\cdot4^{4}\cdot7^{3}\cdot7^{3}\cdot121^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{100999381393}{723148272} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/4\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-8 a : 8 a - 37 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 0.46479157596705282806215681897356740753 \) | ||
| Tamagawa product: | \( 64 \) = \(2^{2}\cdot2^{2}\cdot1\cdot1\cdot2^{2}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.1467803320131867467924918687634274992 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((2)\) | \(4\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((-3a+1)\) | \(7\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((3a-2)\) | \(7\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
| \((11)\) | \(121\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
71148.2-e
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 462.g4 |
| \(\Q\) | 1386.a4 |