Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 13 x + 91 x^{2} - 325 x^{3} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.224314219339$, $\pm0.319105789696$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.74525.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $379$ | $399845$ | $250094899$ | $153368146805$ | $95406034209904$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $639$ | $16003$ | $392619$ | $9769578$ | $244125039$ | $6103365463$ | $152587395699$ | $3814696960633$ | $95367433652974$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+3)x^6+(4a+2)x^5+(a+2)x^4+(4a+4)x^3+(a+2)x+a+2$
- $y^2=(2a+3)x^6+(2a+3)x^5+(2a+1)x^4+x^3+(4a+4)x^2+4x+a+4$
- $y^2=4x^6+4x^5+x^4+(2a+4)x^3+(a+2)x^2+(a+2)x+4$
- $y^2=(4a+3)x^6+3x^5+(a+4)x^4+(3a+4)x^3+(2a+2)x^2+3ax+4a+3$
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 4.0.74525.1. |
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.n_dn | $2$ | 2.625.n_bpp |