Base field 3.3.148.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -1, 1]))
gp: K = nfinit(Polrev([1, -3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-1,1]),K([0,0,0]),K([-1,0,1]),K([238104,-604593,-516708]),K([168997390,-429116293,-366738953])])
gp: E = ellinit([Polrev([-2,-1,1]),Polrev([0,0,0]),Polrev([-1,0,1]),Polrev([238104,-604593,-516708]),Polrev([168997390,-429116293,-366738953])], K);
magma: E := EllipticCurve([K![-2,-1,1],K![0,0,0],K![-1,0,1],K![238104,-604593,-516708],K![168997390,-429116293,-366738953]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+2a+3)\) | = | \((a^2-a-2)^{2}\cdot(a^2-a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 20 \) | = | \(2^{2}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-384a^2+1350a+626)\) | = | \((a^2-a-2)^{4}\cdot(a^2-a-1)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 3906250000 \) | = | \(2^{4}\cdot5^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{3077332109418143868}{244140625} a^{2} + \frac{2119953522630977224}{244140625} a + \frac{9891535601060896996}{244140625} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{1273}{4} a^{2} + 372 a - \frac{589}{4} : -86 a^{2} - \frac{201}{2} a + \frac{157}{4} : 1\right)$ | $\left(-143 a^{2} - 167 a + 66 : 38 a^{2} + 45 a - 17 : 1\right)$ |
| 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: | \( 8.3322759945879467072253395992809508380 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 0.77052247619568588510982188104015429810 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-a-2)\) | \(2\) | \(1\) | \(IV\) | Additive | \(-1\) | \(2\) | \(4\) | \(0\) |
| \((a^2-a-1)\) | \(5\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
20.1-a
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.