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([0,0]),K([0,1]),K([1,0]),K([912,-912]),K([10919,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,1]),Polrev([1,0]),Polrev([912,-912]),Polrev([10919,0])], K);
magma: E := EllipticCurve([K![0,0],K![0,1],K![1,0],K![912,-912],K![10919,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-98a+49)\) | = | \((-2a+1)\cdot(-3a+1)^{2}\cdot(3a-2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 7203 \) | = | \(3\cdot7^{2}\cdot7^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-78121827)\) | = | \((-2a+1)^{26}\cdot(-3a+1)^{2}\cdot(3a-2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6103019853817929 \) | = | \(3^{26}\cdot7^{2}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1713910976512}{1594323} \) | ||
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 | $\left(78 a : -729 a + 364 : 1\right)$ |
| Height | \(0.99151044712942316397777192312828373434\) |
| 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: | \( 0.99151044712942316397777192312828373434 \) | ||
| Period: | \( 0.41695610578783595588572306773421569370 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.9094882578568942878120978565977180232 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(2\) | \(I_{26}\) | Non-split multiplicative | \(1\) | \(1\) | \(26\) | \(26\) |
| \((-3a+1)\) | \(7\) | \(1\) | \(II\) | Additive | \(-1\) | \(2\) | \(2\) | \(0\) |
| \((3a-2)\) | \(7\) | \(1\) | \(II\) | Additive | \(-1\) | \(2\) | \(2\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cn |
| \(13\) | 13B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
13.
Its isogeny class
7203.3-b
consists of curves linked by isogenies of
degree 13.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 147.b1 |
| \(\Q\) | 441.a1 |