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([1,1]),K([0,0]),K([1,1]),K([150,-378]),K([-2246,1815])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([1,1]),Polrev([150,-378]),Polrev([-2246,1815])], K);
magma: E := EllipticCurve([K![1,1],K![0,0],K![1,1],K![150,-378],K![-2246,1815]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((210a-144)\) | = | \((-2a+1)^{2}\cdot(2)\cdot(6a-5)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 34596 \) | = | \(3^{2}\cdot4\cdot31^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-189034236a-397213308)\) | = | \((-2a+1)^{9}\cdot(2)^{2}\cdot(6a-5)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 268599268639219248 \) | = | \(3^{9}\cdot4^{2}\cdot31^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{27357}{4} a + \frac{2559}{2} \) | ||
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 | $\left(2 a - 29 : -165 a + 54 : 1\right)$ |
| Height | \(0.20934117376004539619924328136949969586\) |
| 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: | \( 0.20934117376004539619924328136949969586 \) | ||
| Period: | \( 0.44818003686842133002290338247418375706 \) | ||
| Tamagawa product: | \( 12 \) = \(2\cdot2\cdot3\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.6000863595140171338428741983971979992 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(2\) | \(9\) | \(0\) |
| \((2)\) | \(4\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((6a-5)\) | \(31\) | \(3\) | \(IV^{*}\) | Additive | \(1\) | \(2\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 34596.3-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.