Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([0,4,1,-1]),K([1,0,0,0]),K([-570,1002,274,-220]),K([7022,-12765,-3453,2785])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([0,4,1,-1]),Polrev([1,0,0,0]),Polrev([-570,1002,274,-220]),Polrev([7022,-12765,-3453,2785])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![0,4,1,-1],K![1,0,0,0],K![-570,1002,274,-220],K![7022,-12765,-3453,2785]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+2a+2)\) | = | \((-a^2+2a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 13 \) | = | \(13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3-a^2-4a-2)\) | = | \((-a^2+2a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 13 \) | = | \(13\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \) | ||
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: | \(1\) |
| Generator | $\left(\frac{213}{49} a^{3} - \frac{229}{49} a^{2} - \frac{894}{49} a + \frac{543}{49} : -\frac{603}{343} a^{3} + \frac{963}{343} a^{2} + \frac{2720}{343} a - \frac{2540}{343} : 1\right)$ |
| Height | \(0.83268551475266936289684476785956567701\) |
| 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: | $\left(\frac{13}{3} a^{3} - \frac{13}{3} a^{2} - \frac{52}{3} a + \frac{35}{3} : -\frac{7}{9} a^{3} + \frac{49}{9} a^{2} + \frac{26}{3} a - \frac{26}{3} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.83268551475266936289684476785956567701 \) | ||
| Period: | \( 403.69802529924675885800667326782354074 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 1.50086006684516 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2+2a+2)\) | \(13\) | \(1\) | \(I_{1}\) | 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
13.1-b
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.