Base field 6.6.905177.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 9 x^{3} + 7 x^{2} - 9 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -9, 7, 9, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -9, 7, 9, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 7, 9, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-1,-4,1,1,0]),K([-3,-9,-1,7,0,-1]),K([3,8,-3,-6,1,1]),K([-16,22,19,-21,-2,3]),K([-73,40,73,-47,-9,7])])
gp: E = ellinit([Polrev([1,-1,-4,1,1,0]),Polrev([-3,-9,-1,7,0,-1]),Polrev([3,8,-3,-6,1,1]),Polrev([-16,22,19,-21,-2,3]),Polrev([-73,40,73,-47,-9,7])], K);
magma: E := EllipticCurve([K![1,-1,-4,1,1,0],K![-3,-9,-1,7,0,-1],K![3,8,-3,-6,1,1],K![-16,22,19,-21,-2,3],K![-73,40,73,-47,-9,7]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+7a^3-a^2-8a-3)\) | = | \((-a^5+7a^3-a^2-8a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 43 \) | = | \(43\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((13a^5+5a^4-80a^3+75a+25)\) | = | \((-a^5+7a^3-a^2-8a-3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -3418801 \) | = | \(-43^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{3649320162593979}{3418801} a^{5} + \frac{3305647721589223}{3418801} a^{4} - \frac{19245197647346057}{3418801} a^{3} - \frac{3833979657134919}{3418801} a^{2} + \frac{18238238681182310}{3418801} a + \frac{1914832530089513}{3418801} \) | ||
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: | $\left(\frac{5}{4} a^{5} + a^{4} - \frac{33}{4} a^{3} - \frac{19}{4} a^{2} + \frac{43}{4} a + \frac{15}{2} : \frac{5}{2} a^{5} + \frac{3}{8} a^{4} - \frac{131}{8} a^{3} + 5 a^{2} + \frac{135}{8} a - \frac{11}{4} : 1\right)$ |
| 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: | \( 3725.9292265925486173189383494524689995 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.95811 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+7a^3-a^2-8a-3)\) | \(43\) | \(2\) | \(I_{4}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
43.3-c
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.