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([0,0]),K([0,0]),K([0,1]),K([0,0]),K([2,-1])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,1]),Polrev([0,0]),Polrev([2,-1])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![0,0],K![2,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-360a+63)\) | = | \((-2a+1)^{4}\cdot(-7a+4)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 110889 \) | = | \(3^{4}\cdot37^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((891a-1080)\) | = | \((-2a+1)^{6}\cdot(-7a+4)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 998001 \) | = | \(3^{6}\cdot37^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 0 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | (complex multiplication) | |
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-3})/2]\) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{U}(1)$ | ||
Mordell-Weil group
| Rank: | \(2\) | |
| Generators | $\left(-2 a + 1 : -2 a - 1 : 1\right)$ | $\left(-1 : -a + 1 : 1\right)$ |
| Heights | \(0.95012803481821618733420527593041820609\) | \(0.31670934493940539577806842531013940203\) |
| 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: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.075228606878960452632260504063342903679 \) | ||
| Period: | \( 4.4420192558559418852128181535285937496 \) | ||
| Tamagawa product: | \( 3 \) = \(3\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 4.6303526740063554708484726635220760852 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(3\) | \(IV\) | Additive | \(-1\) | \(4\) | \(6\) | \(0\) |
| \((-7a+4)\) | \(37\) | \(1\) | \(II\) | Additive | \(1\) | \(2\) | \(2\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=3\), a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 110889.1-CMa consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.