Base field 3.3.229.1
Generator \(a\), with minimal polynomial \( x^{3} - 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -4, 0, 1]))
gp: K = nfinit(Polrev([-1, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0]),K([-3,1,1]),K([-3,0,1]),K([-71,327,-138]),K([-603,-1062,637])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([-3,1,1]),Polrev([-3,0,1]),Polrev([-71,327,-138]),Polrev([-603,-1062,637])], K);
magma: E := EllipticCurve([K![1,0,0],K![-3,1,1],K![-3,0,1],K![-71,327,-138],K![-603,-1062,637]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a-3)\) | = | \((a+1)\cdot(-a^2+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-987a^2+166a+12389)\) | = | \((a+1)^{6}\cdot(-a^2+2)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 885842380864 \) | = | \(2^{6}\cdot7^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{144450740091078986229263}{885842380864} a^{2} + \frac{305499959947366052778019}{885842380864} a + \frac{68301210357720562240171}{885842380864} \) | ||
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: | $\left(3 a^{2} - 2 a - 11 : -2 a^{2} + a + 7 : 1\right)$ | $\left(a^{2} - 9 a + \frac{51}{4} : -a^{2} + \frac{9}{2} a - \frac{39}{8} : 1\right)$ |
| 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: | \( 6.3395132473657773066417021970917495623 \) | ||
| Tamagawa product: | \( 24 \) = \(2\cdot( 2^{2} \cdot 3 )\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 0.62839024075361314906521287939860880115 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((-a^2+2)\) | \(7\) | \(12\) | \(I_{12}\) | 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
14.1-b
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.