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([-1,-1]),K([0,0]),K([143,2]),K([-463,-705])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([143,2]),Polrev([-463,-705])], K);
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![143,2],K![-463,-705]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-432a+144)\) | = | \((-2a+1)^{4}\cdot(2)^{4}\cdot(-3a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 145152 \) | = | \(3^{4}\cdot4^{4}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-425926656a-75644928)\) | = | \((-2a+1)^{4}\cdot(2)^{13}\cdot(-3a+1)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 219354862649868288 \) | = | \(3^{4}\cdot4^{13}\cdot7^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{50481832659}{80707214} a + \frac{60742004649}{80707214} \) | ||
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 | $\left(-16 a + 1 : -24 a + 64 : 1\right)$ |
| Height | \(0.11836964935810119256124156829383129363\) |
| 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: | \( 0.11836964935810119256124156829383129363 \) | ||
| Period: | \( 0.49542648343354191392605963966791031638 \) | ||
| Tamagawa product: | \( 36 \) = \(1\cdot2^{2}\cdot3^{2}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 4.8755256354773743747121083913080549467 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(1\) | \(II\) | Additive | \(1\) | \(4\) | \(4\) | \(0\) |
| \((2)\) | \(4\) | \(4\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
| \((-3a+1)\) | \(7\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
145152.1-f
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.