Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 11 x + 61 x^{2} - 176 x^{3} + 256 x^{4}$ |
Frobenius angles: | $\pm0.189901625224$, $\pm0.315486115946$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.22625.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $131$ | $66155$ | $17419856$ | $4342480355$ | $1101116001591$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $258$ | $4251$ | $66258$ | $1050106$ | $16776903$ | $268426206$ | $4294969218$ | $68719615491$ | $1099511489098$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+1)x^6+(a^2+a)x^5+(a^2+a)x^4+(a+1)x^3+a^2x^2+(a^3+a^2+1)x+a$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2)x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a^2+1)x^3+(a^3+1)x^2+(a^3+a^2+a)x+a^3+a^2+a$
- $y^2+(x^3+ax+a)y=(a^3+a)x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+a^2x^3+(a^3+a^2+a)x^2+(a^3+1)x+a^3+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2)x^6+(a^2+a)x^5+(a^2+a)x^4+ax^3+(a^3+a+1)x^2+(a^3+a+1)x+a^3+a^2+1$
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 4.0.22625.1. |
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.l_cj | $2$ | 2.256.b_nx |