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([0,0]),K([-1,0]),K([1,0]),K([-7,0]),K([10,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-7,0]),Polrev([10,0])], K);
magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-7,0],K![10,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((11)\) | = | \((-2a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 121 \) | = | \(11^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1331)\) | = | \((-2a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1771561 \) | = | \(11^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -32768 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z[(1+\sqrt{-11})/2]\) | (complex multiplication) | |
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-11})/2]\) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{U}(1)$ | ||
Mordell-Weil group
| Rank: | \(2\) | |
| Generators | $\left(-\frac{2}{9} a + \frac{5}{3} : -\frac{11}{27} a - \frac{10}{9} : 1\right)$ | $\left(4 : 5 : 1\right)$ |
| Heights | \(0.44892578080345226663151545082504359911\) | \(0.089785156160690453326303090165008719822\) |
| 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: | \( 0.022168779233698819878879540118695732679 \) | ||
| Period: | \( 3.4769158424352161557941951589028318460 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.74368597809907052242861130738004104123 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(11\) | \(4\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(11\) | 11B.1.5[2] |
For all other primes \(p\), the image is a split Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 121.1-CMa consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 121.b1 |
| \(\Q\) | 121.b2 |