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([0,1]),K([0,0]),K([1,1]),K([-9186,5302]),K([-975566,563242])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([1,1]),Polrev([-9186,5302]),Polrev([-975566,563242])], K);
magma: E := EllipticCurve([K![0,1],K![0,0],K![1,1],K![-9186,5302],K![-975566,563242]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((27a+27)\) | = | \((a+1)\cdot(a)^{6}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 1458 \) | = | \(2\cdot3^{6}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-169869312)\) | = | \((a+1)^{42}\cdot(a)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 28855583159353344 \) | = | \(2^{42}\cdot3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1159088625}{2097152} \) | ||
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 | $\left(-51 a + 91 : -258 a + 448 : 1\right)$ |
| Height | \(1.0267244893349748108615965951594791037\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-75 a + 131 : 894 a - 1552 : 1\right)$ |
| 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: | \( 1.0267244893349748108615965951594791037 \) | ||
| Period: | \( 0.52085241842792041319755632697253449779 \) | ||
| Tamagawa product: | \( 126 \) = \(( 2 \cdot 3 \cdot 7 )\cdot3\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 4.3225100752781365183402746128135951273 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(2\) | \(42\) | \(I_{42}\) | Split multiplicative | \(-1\) | \(1\) | \(42\) | \(42\) |
| \((a)\) | \(3\) | \(3\) | \(IV\) | Additive | \(1\) | \(6\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
| \(7\) | 7B.2.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 7 and 21.
Its isogeny class
1458.1-p
consists of curves linked by isogenies of
degrees dividing 21.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 162.c2 |
| \(\Q\) | 1296.f2 |