Invariants
Base field: | $\F_{3^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 28 x + 351 x^{2} - 2268 x^{3} + 6561 x^{4}$ |
Frobenius angles: | $\pm0.124262281589$, $\pm0.282730282950$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.88592.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4617$ | $42517953$ | $282816754308$ | $1853532787754313$ | $12157928970691398057$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $54$ | $6480$ | $532170$ | $43058628$ | $3486859974$ | $282429662598$ | $22876791686838$ | $1853020216359684$ | $150094636127033418$ | $12157665470073084240$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=a^2x^6+(a^3+a^2+1)x^4+(2a^3+a^2+a)x^3+(a^2+a)x+2a$
- $y^2=(a^3+2a+1)x^6+(a^2+2a)x^4+(a^3+2a+1)x^3+(a^3+2a^2+a+1)x+a^3+2a+2$
- $y^2=(a^3+a^2+a+1)x^6+(a^2+a+1)x^4+a^3x^3+(a^3+a^2+a)x+a+2$
- $y^2=2x^6+(2a^2+a+2)x^4+(a^3+2a^2+2a)x^3+(a^3+2a+2)x+a^2+2a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3^{4}}$The endomorphism algebra of this simple isogeny class is 4.0.88592.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.81.bc_nn | $2$ | (not in LMFDB) |