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([-9,450]),K([-3321,81])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,1]),Polrev([0,0]),Polrev([-9,450]),Polrev([-3321,81])], K);
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![-9,450],K![-3321,81]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-240a+96)\) | = | \((-2a+1)^{2}\cdot(2)^{4}\cdot(-5a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 43776 \) | = | \(3^{2}\cdot4^{4}\cdot19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-268738560a+1128701952)\) | = | \((-2a+1)^{16}\cdot(2)^{13}\cdot(-5a+2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1042862772829814784 \) | = | \(3^{16}\cdot4^{13}\cdot19^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{537398275}{175446} a + \frac{667275508}{87723} \) | ||
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 | $\left(18 a + 9 : 162 a : 1\right)$ |
| Height | \(0.99303087123390914188810305386629033404\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-9 a + 9 : 0 : 1\right)$ |
| 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: | \( 0.99303087123390914188810305386629033404 \) | ||
| Period: | \( 0.39564600915738189826274701798960759357 \) | ||
| Tamagawa product: | \( 16 \) = \(2^{2}\cdot2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.6293503581478549821045159025949889015 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(4\) | \(I_{10}^{*}\) | Additive | \(-1\) | \(2\) | \(16\) | \(10\) |
| \((2)\) | \(4\) | \(2\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
| \((-5a+2)\) | \(19\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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 |
| \(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
43776.2-l
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.