Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0,0,0,0]),K([4,-9,-11,9,3,-2]),K([1,-2,-6,4,2,-1]),K([1,12,3,-7,-1,1]),K([7,-17,-15,12,4,-2])])
gp: E = ellinit([Polrev([0,0,0,0,0,0]),Polrev([4,-9,-11,9,3,-2]),Polrev([1,-2,-6,4,2,-1]),Polrev([1,12,3,-7,-1,1]),Polrev([7,-17,-15,12,4,-2])], K);
magma: E := EllipticCurve([K![0,0,0,0,0,0],K![4,-9,-11,9,3,-2],K![1,-2,-6,4,2,-1],K![1,12,3,-7,-1,1],K![7,-17,-15,12,4,-2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4+a^3-3a^2-3a+2)\) | = | \((a^4+a^3-3a^2-3a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 79 \) | = | \(79\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^4+2a^3+3a^2-5a)\) | = | \((a^4+a^3-3a^2-3a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -79 \) | = | \(-79\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{442196209664}{79} a^{5} - \frac{139077779456}{79} a^{4} - \frac{2003217559552}{79} a^{3} + \frac{161219244032}{79} a^{2} + \frac{1156138635264}{79} a - \frac{262359261184}{79} \) | ||
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 | $\left(3 a^{5} - 3 a^{4} - 12 a^{3} + 11 a^{2} + 8 a - 3 : 6 a^{5} - a^{4} - 26 a^{3} + 6 a^{2} + 23 a - 6 : 1\right)$ |
| Height | \(0.0039643022893652606001291707014763822084\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.0039643022893652606001291707014763822084 \) | ||
| Period: | \( 73613.057358515183571431418106340574291 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.51389 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4+a^3-3a^2-3a+2)\) | \(79\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 79.1-e consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.