Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 25 x^{2} - 63 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.0842035494981$, $\pm0.435433986784$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.37485.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $37$ | $6549$ | $526621$ | $41815365$ | $3455537152$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $83$ | $723$ | $6371$ | $58518$ | $531971$ | $4788507$ | $43054211$ | $387410907$ | $3486836078$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2ax^6+2x^5+ax^4+(a+1)x^3+(a+1)x^2+x+2a$
- $y^2=(a+2)x^6+x^5+x^4+(a+1)x^3+(a+1)x^2+(a+2)x+2a+1$
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 4.0.37485.2. |
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_z | $2$ | 2.81.b_adr |