Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 36 x^{2} - 112 x^{3} + 256 x^{4}$ |
Frobenius angles: | $\pm0.206663904003$, $\pm0.474998167614$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.101277.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $174$ | $71688$ | $17102286$ | $4291673808$ | $1100825995494$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $280$ | $4174$ | $65488$ | $1049830$ | $16790488$ | $268458046$ | $4294800544$ | $68718661846$ | $1099511059240$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=ax^5+(a^3+a^2)x^3+(a^3+a^2)x^2+x$
- $y^2+xy=a^2x^5+(a^3+a^2+a+1)x^3+(a^3+1)x^2+x$
- $y^2+xy=(a+1)x^5+(a^3+a)x^3+(a^3+1)x^2+x$
- $y^2+xy=(a^3+a+1)x^5+a^3x^3+(a^3+1)x^2+x$
- $y^2+xy=(a^2+1)x^5+a^3x^3+(a^3+a+1)x^2+x$
- $y^2+xy=(a^3+a^2+a)x^5+(a^3+a)x^3+(a^3+1)x^2+x$
- $y^2+xy=(a^3+a^2+1)x^5+(a^3+a^2+a+1)x^3+(a^3+a^2)x^2+x$
- $y^2+xy=(a^3+1)x^5+(a^3+a^2)x^3+(a^3+a^2+1)x^2+x$
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 4.0.101277.1. |
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_bk | $2$ | 2.256.x_jg |