Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 91 x^{2} - 350 x^{3} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.0590489725365$, $\pm0.363026417733$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.17984.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 3 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $353$ | $381593$ | $244558400$ | $152301779753$ | $95279834032353$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $612$ | $15654$ | $389892$ | $9756652$ | $244100334$ | $6103493772$ | $152588569092$ | $3814700259414$ | $95367429346852$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(3a+2)x^6+2ax^5+(4a+2)x^4+x^3+(4a+3)x^2+4x+4a+4$
- $y^2=2x^6+(4a+3)x^5+(4a+4)x^4+2ax^3+(a+4)x^2+(3a+4)x+a+4$
- $y^2=3ax^6+(a+4)x^5+(3a+1)x^4+(4a+4)x^3+(4a+1)x^2+(a+2)x+2a+4$
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 4.0.17984.1. |
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.o_dn | $2$ | 2.625.ao_akj |