Base field 4.4.16357.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -6, 0, 1]))
gp: K = nfinit(Polrev([1, -1, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-5,1,1]),K([3,-1,-1,0]),K([-3,-5,1,1]),K([-9,-26,-2,4]),K([-68,-137,6,22])])
gp: E = ellinit([Polrev([-3,-5,1,1]),Polrev([3,-1,-1,0]),Polrev([-3,-5,1,1]),Polrev([-9,-26,-2,4]),Polrev([-68,-137,6,22])], K);
magma: E := EllipticCurve([K![-3,-5,1,1],K![3,-1,-1,0],K![-3,-5,1,1],K![-9,-26,-2,4],K![-68,-137,6,22]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+6a-1)\) | = | \((-a-1)\cdot(a^2+2a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a^3+a^2-32a+2)\) | = | \((-a-1)^{6}\cdot(a^2+2a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 18225 \) | = | \(3^{6}\cdot5^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{141544361317964}{18225} a^{3} + \frac{328404077969408}{18225} a^{2} + \frac{87307968145483}{18225} a - \frac{61000042547837}{18225} \) | ||
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 | $\left(a^{3} + a^{2} - 5 a - 4 : 2 a^{3} - 12 a - 6 : 1\right)$ |
| Height | \(0.092267086698763665584567795674922022853\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{3}{4} a^{3} + \frac{1}{2} a^{2} - \frac{21}{4} a - 4 : \frac{19}{8} a^{3} + \frac{7}{8} a^{2} - \frac{51}{4} a - \frac{13}{2} : 1\right)$ |
| 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: | \( 0.092267086698763665584567795674922022853 \) | ||
| Period: | \( 459.40752215200069890860679692984019452 \) | ||
| Tamagawa product: | \( 12 \) = \(( 2 \cdot 3 )\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.97717159681373 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a-1)\) | \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((a^2+2a)\) | \(5\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
15.2-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.