Base field \(\Q(\sqrt{6}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 6 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
gp: K = nfinit(Polrev([-6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,1]),K([-3,0]),K([241,99])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-3,0]),Polrev([241,99])], K);
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,1],K![-3,0],K![241,99]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((28)\) | = | \((-a+2)^{4}\cdot(7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 784 \) | = | \(2^{4}\cdot49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-112)\) | = | \((-a+2)^{8}\cdot(7)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 12544 \) | = | \(2^{8}\cdot49\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{4}{7} \) | ||
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: | \(2\) | |
| Generators | $\left(18 a + 44 : -196 a - 479 : 1\right)$ | $\left(0 : 4 a + 11 : 1\right)$ |
| Heights | \(1.2934556644303437231429396362392536026\) | \(0.23991989872520257601163977124532260608\) |
| 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: | $\left(-2 a - 5 : 2 a + 6 : 1\right)$ | |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.25276419421135621330510818787154500375 \) | ||
| Period: | \( 22.757121047190831995759190658358911470 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.3483198413101714783917097596769912591 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(4\) | \(8\) | \(0\) |
| \((7)\) | \(49\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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.
Its isogeny class
784.1-n
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 112.a2 |
| \(\Q\) | 4032.d2 |