Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 93 x^{2} - 350 x^{3} + 625 x^{4}$ |
| Frobenius angles: | $\pm0.106111050681$, $\pm0.349621014660$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.489024.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $355$ | $384465$ | $245888620$ | $152626838025$ | $95330397755275$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $616$ | $15738$ | $390724$ | $9761832$ | $244123846$ | $6103597848$ | $152589213124$ | $3814704632202$ | $95367451045576$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+4)x^6+(2a+4)x^5+(a+3)x^4+(4a+4)x+3a+2$
- $y^2=2ax^6+(3a+3)x^5+4x^4+3x+2a+4$
- $y^2=(2a+2)x^6+(a+1)x^5+(4a+2)x^4+(3a+3)x+2a$
- $y^2=(3a+3)x^6+(4a+2)x^5+(a+3)x^4+2x+3a+2$
- $y^2=(2a+4)x^6+2ax^5+ax^4+(4a+1)x^3+(4a+4)x^2+(2a+1)x+4a+3$
- $y^2=(a+3)x^6+(2a+3)x^5+3x^4+(4a+1)x^3+2x^2+4x+4a+3$
- $y^2=(3a+2)x^6+2ax^5+(a+2)x^4+4x^3+4x^2+(4a+3)x+a+4$
- $y^2=(2a+1)x^6+2ax^5+3x^4+(4a+1)x^3+(a+2)x^2+4$
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 4.0.489024.2. |
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_dp | $2$ | 2.625.ak_dv |