Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x + 101 x^{2} - 375 x^{3} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.0651474250922$, $\pm0.325607171919$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.132741.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $337$ | $376429$ | $244824097$ | $152538441525$ | $95311795063552$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $603$ | $15671$ | $390499$ | $9759926$ | $244099203$ | $6103394231$ | $152588195299$ | $3814702125491$ | $95367454132878$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+3)x^6+(2a+3)x^5+(2a+4)x^4+(3a+4)x^3+(2a+3)x^2+(4a+2)x+4a+1$
- $y^2=(3a+2)x^6+(2a+3)x^5+(4a+4)x^4+2ax^3+(3a+1)x^2+2x+4a+3$
- $y^2=(4a+3)x^6+(2a+3)x^5+(3a+4)x^4+(a+1)x^3+3x^2+(3a+1)x$
- $y^2=(a+2)x^6+(3a+4)x^5+(4a+3)x^4+(2a+1)x^3+(4a+4)x^2+2a+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$| The endomorphism algebra of this simple isogeny class is 4.0.132741.1. |
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 |
|---|---|---|
| 2.25.p_dx | $2$ | 2.625.ax_ht |