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([0,1]),K([0,1]),K([1,1]),K([-924,1233]),K([2145,-15358])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,1]),Polrev([1,1]),Polrev([-924,1233]),Polrev([2145,-15358])], K);
magma: E := EllipticCurve([K![0,1],K![0,1],K![1,1],K![-924,1233],K![2145,-15358]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((150a-36)\) | = | \((-2a+1)^{2}\cdot(2)\cdot(-3a+1)\cdot(9a-8)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 18396 \) | = | \(3^{2}\cdot4\cdot7\cdot73\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-133879392a-769824)\) | = | \((-2a+1)^{15}\cdot(2)^{5}\cdot(-3a+1)^{5}\cdot(9a-8)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 18027347800347648 \) | = | \(3^{15}\cdot4^{5}\cdot7^{5}\cdot73\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{12211771579546037}{9540459936} a + \frac{983887313721709}{2385114984} \) | ||
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 | $\left(-11 a + 18 : 23 a - 33 : 1\right)$ |
| Height | \(0.17178873498510995297490819415192030080\) |
| 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: | \( 0.17178873498510995297490819415192030080 \) | ||
| Period: | \( 0.37914079994429934286599921290818079769 \) | ||
| Tamagawa product: | \( 20 \) = \(2^{2}\cdot5\cdot1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.0083236874601331375272706309916027120 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(4\) | \(I_{9}^{*}\) | Additive | \(-1\) | \(2\) | \(15\) | \(9\) |
| \((2)\) | \(4\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((-3a+1)\) | \(7\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
| \((9a-8)\) | \(73\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 18396.2-d 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.