Base field 4.4.10889.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([1, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-6,0,1]),K([-2,4,1,-1]),K([2,-4,-1,1]),K([-12,22,6,-5]),K([19,-22,-10,6])])
gp: E = ellinit([Polrev([0,-6,0,1]),Polrev([-2,4,1,-1]),Polrev([2,-4,-1,1]),Polrev([-12,22,6,-5]),Polrev([19,-22,-10,6])], K);
magma: E := EllipticCurve([K![0,-6,0,1],K![-2,4,1,-1],K![2,-4,-1,1],K![-12,22,6,-5],K![19,-22,-10,6]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2-2a+1)\) | = | \((-a^3+a^2+4a)\cdot(-a+2)\) |
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: | \((7a^3+6a^2-66a+3)\) | = | \((-a^3+a^2+4a)^{8}\cdot(-a+2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 614656 \) | = | \(2^{8}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{10156308195}{614656} a^{3} + \frac{1932332501}{307328} a^{2} + \frac{53871409229}{614656} a + \frac{16864249233}{614656} \) | ||
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 | $\left(-4 a^{3} + 18 a^{2} - 22 a + 6 : -69 a^{3} + 217 a^{2} - 95 a - 16 : 1\right)$ | |
| Height | \(0.55480061553011371641761631483700325079\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(a^{3} - \frac{5}{4} a^{2} - \frac{7}{2} a + \frac{3}{4} : -\frac{5}{8} a^{3} + \frac{17}{8} a^{2} - \frac{5}{8} a - \frac{27}{8} : 1\right)$ | $\left(a^{2} - 2 a - 1 : a^{3} - 8 a + 2 : 1\right)$ |
| 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: | \( 0.55480061553011371641761631483700325079 \) | ||
| Period: | \( 720.45198737899303252606251929718486329 \) | ||
| Tamagawa product: | \( 16 \) = \(2^{3}\cdot2\) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 3.83043395919801 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a)\) | \(2\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((-a+2)\) | \(7\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
14.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.