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([0,0]),K([0,0]),K([6,0]),K([-28,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([6,0]),Polrev([-28,0])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![6,0],K![-28,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((178)\) | = | \((2)\cdot(5a+2)\cdot(5a-7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 31684 \) | = | \(4\cdot89\cdot89\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-364544)\) | = | \((2)^{12}\cdot(5a+2)\cdot(5a-7)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 132892327936 \) | = | \(4^{12}\cdot89\cdot89\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{23639903}{364544} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(4 : -10 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 1.6559463612394139590793461865527010610 \) | ||
| Tamagawa product: | \( 12 \) = \(( 2^{2} \cdot 3 )\cdot1\cdot1\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 5.3257238818768082896305597675588423420 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
| \((5a+2)\) | \(89\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((5a-7)\) | \(89\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
31684.2-c
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 |
|---|---|
| \(\Q\) | 178.b2 |
| \(\Q\) | 21538.b2 |