Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 28 x^{2} - 112 x^{3} + 256 x^{4}$ |
Frobenius angles: | $\pm0.109519428602$, $\pm0.521148502204$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.122525.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $166$ | $67064$ | $16414246$ | $4256283824$ | $1100534194126$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $264$ | $4006$ | $64944$ | $1049550$ | $16788648$ | $268445110$ | $4294975584$ | $68720295646$ | $1099515282904$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=(a^3+a+1)x^5+(a^3+a^2)x^3+a^3x^2+x$
- $y^2+xy=(a^3+a^2+a)x^5+a^3x^3+(a^3+a^2+1)x^2+x$
- $y^2+xy=(a^3+a^2+1)x^5+(a^3+a)x^3+(a^3+a)x^2+x$
- $y^2+xy=(a^3+1)x^5+(a^3+a^2+a+1)x^3+(a^3+a)x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$The endomorphism algebra of this simple isogeny class is 4.0.122525.1. |
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 |
---|---|---|
2.16.h_bc | $2$ | 2.256.h_akm |