Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 29 x^{2} - 63 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.220419591014$, $\pm0.370053256546$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.11125.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $41$ | $7421$ | $590441$ | $44117845$ | $3488356096$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $91$ | $807$ | $6723$ | $59078$ | $531091$ | $4783607$ | $43050083$ | $387401163$ | $3486620206$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2ax^6+(a+2)x^5+(a+1)x^4+2x^3+2x^2+x$
- $y^2=(2a+1)x^6+x^5+x^4+(2a+2)x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.11125.1. |
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 |
---|---|---|
2.9.h_bd | $2$ | 2.81.j_er |