Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x + 13 x^{2} )( 1 + 7 x + 13 x^{2} )$ |
| $1 + 2 x - 9 x^{2} + 26 x^{3} + 169 x^{4}$ | |
| Frobenius angles: | $\pm0.256122854178$, $\pm0.922789520844$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $4$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $189$ | $25137$ | $5143824$ | $819893529$ | $138305860749$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $148$ | $2338$ | $28708$ | $372496$ | $4825798$ | $62733904$ | $815694916$ | $10604262634$ | $137859194068$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=x^6+9$
- $y^2=x^6+5 x^5+8 x^4+11 x^3+4 x^2+6 x+7$
- $y^2=x^6+x^3+3$
- $y^2=10 x^6+8 x^5+9 x^4+6 x^3+x^2+7 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{3}}$.
Endomorphism algebra over $\F_{13}$| The isogeny class factors as 1.13.af $\times$ 1.13.h and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{13^{3}}$ is 1.2197.cs 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.