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([1,0]),K([-1,0]),K([0,0]),K([-264,15]),K([414,-681])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([-264,15]),Polrev([414,-681])], K);
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-264,15],K![414,-681]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-15a+30)\) | = | \((-a+2)\cdot(a+3)^{2}\cdot(-a-1)\cdot(-a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 450 \) | = | \(2\cdot3^{2}\cdot5\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1306125a+7398000)\) | = | \((-a+2)\cdot(a+3)^{6}\cdot(-a-1)^{12}\cdot(-a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 44494628906250 \) | = | \(2\cdot3^{6}\cdot5^{12}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{59305689907281}{488281250} a + \frac{102611946982248}{244140625} \) | ||
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 | $\left(-\frac{9}{2} a + \frac{19}{6} : -\frac{89}{36} a + \frac{29}{3} : 1\right)$ |
| Height | \(1.0972032258924404399265900442888737398\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-7 a + 9 : 26 a - 57 : 1\right)$ |
| 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: | \( 1.0972032258924404399265900442888737398 \) | ||
| Period: | \( 3.0256410151763374151439503775473981520 \) | ||
| Tamagawa product: | \( 72 \) = \(1\cdot2\cdot( 2^{2} \cdot 3 )\cdot3\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.7105588762106606783795993315082418432 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((a+3)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((-a-1)\) | \(5\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
| \((-a+1)\) | \(5\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
450.1-d
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.