Base field 5.5.186037.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 6 x^{3} + 2 x^{2} + 5 x - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 5, 2, -6, -1, 1]))
gp: K = nfinit(Polrev([-2, 5, 2, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 5, 2, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0,0]),K([-2,4,1,-1,0]),K([1,1,0,0,0]),K([25,-47,-38,12,-10]),K([216,-469,-268,363,75])])
gp: E = ellinit([Polrev([1,0,0,0,0]),Polrev([-2,4,1,-1,0]),Polrev([1,1,0,0,0]),Polrev([25,-47,-38,12,-10]),Polrev([216,-469,-268,363,75])], K);
magma: E := EllipticCurve([K![1,0,0,0,0],K![-2,4,1,-1,0],K![1,1,0,0,0],K![25,-47,-38,12,-10],K![216,-469,-268,363,75]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-2)\) | = | \((a^4-a^3-5a^2+2)\cdot(-a^4+a^3+6a^2-a-5)\) |
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: | \((-66a^4+13a^3+433a^2+162a-304)\) | = | \((a^4-a^3-5a^2+2)^{6}\cdot(-a^4+a^3+6a^2-a-5)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -368947264 \) | = | \(-2^{6}\cdot7^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{64477514513737305584321773}{368947264} a^{4} + \frac{10699491548325379494199235}{368947264} a^{3} + \frac{98947272401536641312835617}{92236816} a^{2} + \frac{100578124733041654403631191}{184473632} a - \frac{154611480625383330655439525}{368947264} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a^{3} - 5 a^{2} - 6 a + \frac{15}{4} : -\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - \frac{19}{8} : 1\right)$ |
| 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: | \( 180.84705369920124540685061430705478012 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot2^{3}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.67714945 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-a^3-5a^2+2)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((-a^4+a^3+6a^2-a-5)\) | \(7\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
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.