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([1,0,-3,0,1,0]),K([1,-3,-6,4,2,-1]),K([-1,2,3,-4,-1,1]),K([5,-63,-29,51,-8,1]),K([-73,152,398,-176,-261,120])])
gp: E = ellinit([Polrev([1,0,-3,0,1,0]),Polrev([1,-3,-6,4,2,-1]),Polrev([-1,2,3,-4,-1,1]),Polrev([5,-63,-29,51,-8,1]),Polrev([-73,152,398,-176,-261,120])], K);
magma: E := EllipticCurve([K![1,0,-3,0,1,0],K![1,-3,-6,4,2,-1],K![-1,2,3,-4,-1,1],K![5,-63,-29,51,-8,1],K![-73,152,398,-176,-261,120]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-2)\) | = | \((a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((373a^5-134a^4-1806a^3+34a^2+1898a+342)\) | = | \((a^2-2)^{18}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -150094635296999121 \) | = | \(-9^{18}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{19534593968251450184}{387420489} a^{5} - \frac{25732253879599264147}{387420489} a^{4} - \frac{95706620059011517651}{387420489} a^{3} + \frac{90934603429118494397}{387420489} a^{2} + \frac{101153218100845110134}{387420489} a - \frac{1059714138202501808}{14348907} \) | ||
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: | $\left(-\frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{7}{4} a^{2} + 3 a + \frac{13}{4} : \frac{13}{8} a^{5} + \frac{1}{8} a^{4} - \frac{15}{2} a^{3} + \frac{5}{2} a^{2} + \frac{47}{8} a - \frac{23}{8} : 1\right)$ |
| 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: | \( 1355.3227225432099638829848010340318135 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.972940 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-2)\) | \(9\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
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
9.1-a
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.