Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 13 x + 82 x^{2} - 325 x^{3} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.0779611327259$, $\pm0.393003920309$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.880844.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $370$ | $387020$ | $244538920$ | $152207225600$ | $95286087384850$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $621$ | $15652$ | $389649$ | $9757293$ | $244123986$ | $6103648837$ | $152588889249$ | $3814699124548$ | $95367427613101$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+(a+2)x^5+3x^4+(3a+2)x^3+(4a+3)x^2+(3a+1)x+a+2$
- $y^2=(4a+3)x^6+3x^5+(4a+4)x^4+(a+3)x^3+(3a+3)x^2+(4a+4)x$
- $y^2=(2a+3)x^6+(2a+2)x^5+(a+1)x^4+(a+2)x^3+(3a+4)x^2+(4a+3)x+3a+2$
- $y^2=(2a+4)x^6+(a+1)x^5+(2a+4)x^4+4x^3+(2a+2)x^2+(4a+3)x+2$
- $y^2=(2a+3)x^6+(3a+3)x^5+2ax^3+2x^2+(3a+2)x+4a$
- $y^2=4ax^6+3ax^5+(3a+3)x^4+4x^3+(3a+4)x^2+(3a+3)x+2a+3$
- $y^2=(3a+2)x^6+(3a+4)x^5+(a+2)x^4+2x^3+(2a+4)x+4$
- $y^2=(a+4)x^6+(2a+3)x^5+(2a+1)x^4+4x^3+(2a+2)x^2+(2a+1)x+3a$
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 4.0.880844.1. |
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_de | $2$ | 2.625.af_asi |