Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([39,0]),K([-19,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([39,0]),Polrev([-19,0])], K);
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![39,0],K![-19,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((162)\) | = | \((-a)^{4}\cdot(a-1)^{4}\cdot(2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 26244 \) | = | \(3^{4}\cdot3^{4}\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3779136)\) | = | \((-a)^{10}\cdot(a-1)^{10}\cdot(2)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 14281868906496 \) | = | \(3^{10}\cdot3^{10}\cdot4^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{109503}{64} \) | ||
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: | \(2\) | |
| Generators | $\left(10 : -41 : 1\right)$ | $\left(-8 : 18 a - 5 : 1\right)$ |
| Heights | \(0.10197829462292110456679008894485744001\) | \(0.46540767515234654924340731134483364767\) |
| 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: | \( 0.047461481016454754268391206598584446764 \) | ||
| Period: | \( 1.1018611264310826782954406120478112478 \) | ||
| Tamagawa product: | \( 54 \) = \(3\cdot3\cdot( 2 \cdot 3 )\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 6.8117006149033037391336263069045156559 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(3\) | \(IV^{*}\) | Additive | \(1\) | \(4\) | \(10\) | \(0\) |
| \((a-1)\) | \(3\) | \(3\) | \(IV^{*}\) | Additive | \(1\) | \(4\) | \(10\) | \(0\) |
| \((2)\) | \(4\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
26244.5-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 162.a2 |
| \(\Q\) | 19602.s2 |