Base field \(\Q(\sqrt{3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
gp: K = nfinit(Polrev([-3, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([0,0]),K([-2689,-1538]),K([74761,43117])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([0,0]),Polrev([-2689,-1538]),Polrev([74761,43117])], K);
magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,0],K![-2689,-1538],K![74761,43117]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((16a)\) | = | \((a+1)^{8}\cdot(a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 768 \) | = | \(2^{8}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((18432)\) | = | \((a+1)^{22}\cdot(a)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 339738624 \) | = | \(2^{22}\cdot3^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{3065617154}{9} \) | ||
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: | $\left(10 a + 21 : -12 a - 20 : 1\right)$ |
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: | \( 11.367017035003173802074295255886867379 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.6406875862605360264830853899829179811 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a+1)\) | \(2\) | \(4\) | \(I_{10}^{*}\) | Additive | \(1\) | \(8\) | \(22\) | \(0\) |
\((a)\) | \(3\) | \(2\) | \(I_{4}\) | Non-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, 4, 8 and 16.
Its isogeny class
768.1-e
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 192.b1 |
\(\Q\) | 576.d1 |