Invariants
| Base field: | $\F_{2^{5}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 13 x + 88 x^{2} - 416 x^{3} + 1024 x^{4}$ |
| Frobenius angles: | $\pm0.0988992451558$, $\pm0.436903129149$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.490268.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $684$ | $1054728$ | $1073337588$ | $1097315807184$ | $1125568021887324$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $1032$ | $32756$ | $1046480$ | $33544540$ | $1073785176$ | $34360313476$ | $1099513321120$ | $35184371657324$ | $1125899947071912$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=ax^5+(a^2+a+1)x^3+x$
- $y^2+xy=a^2x^5+(a^4+a^2+1)x^3+x$
- $y^2+xy=a^3x^5+(a^4+a^3+a^2)x^3+x$
- $y^2+xy=a^4x^5+(a^4+a^3+a^2)x^3+x$
- $y^2+xy=(a^3+a)x^5+(a^4+a^2+a+1)x^3+x$
- $y^2+xy=(a^3+a^2+1)x^5+(a^4+a^2+a+1)x^3+x$
- $y^2+xy=(a^3+a^2+a)x^5+(a^4+a^3)x^3+x$
- $y^2+xy=(a^4+a^3+a+1)x^5+(a^4+a^3)x^3+x$
- $y^2+xy=(a^4+a+1)x^5+(a^4+a^2+1)x^3+x$
- $y^2+xy=(a^4+a^3+a^2+a)x^5+(a^2+a+1)x^3+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{5}}$.
Endomorphism algebra over $\F_{2^{5}}$| The endomorphism algebra of this simple isogeny class is 4.0.490268.1. |
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 |
|---|---|---|
| 2.32.n_dk | $2$ | 2.1024.h_abnk |