Base field 3.3.321.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-1,1]),K([3,1,-1]),K([-3,-1,1]),K([-22883,105008,-38894]),K([4932888,-22633861,8383982])])
gp: E = ellinit([Polrev([-2,-1,1]),Polrev([3,1,-1]),Polrev([-3,-1,1]),Polrev([-22883,105008,-38894]),Polrev([4932888,-22633861,8383982])], K);
magma: E := EllipticCurve([K![-2,-1,1],K![3,1,-1],K![-3,-1,1],K![-22883,105008,-38894],K![4932888,-22633861,8383982]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^2-a+2)\) | = | \((-2a^2-a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 43 \) | = | \(43\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-2a^2-a+2)\) | = | \((-2a^2-a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( -43 \) | = | \(-43\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{40941705441282487}{43} a^{2} + \frac{110527380447682623}{43} a - \frac{24088625198885706}{43} \) | ||
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: | $\left(\frac{125}{3} a^{2} - \frac{320}{3} a + \frac{65}{3} : -\frac{443}{9} a^{2} + \frac{434}{3} a - \frac{286}{9} : 1\right)$ |
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: | \( 70.238906800752735083925280484566819961 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(3\) | ||
Leading coefficient: | \( 0.43559483140023977559591703802921499529 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a^2-a+2)\) | \(43\) | \(1\) | \(I_{1}\) | 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 and 9.
Its isogeny class
43.1-a
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.