Invariants
| Base field: | $\F_{13^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 13 x )^{4}$ |
| $1 - 52 x + 1014 x^{2} - 8788 x^{3} + 28561 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $0$, $0$ |
| Angle rank: | $0$ (numerical) |
| Jacobians: | $3$ |
This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $20736$ | $796594176$ | $23255696077056$ | $665323421736960000$ | $19004759032135892541696$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $118$ | $27886$ | $4818022$ | $815616478$ | $137857006678$ | $23298065815246$ | $3937376134705222$ | $665416605920256958$ | $112455406909539395638$ | $19004963774329365471406$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^5+12x$
- $y^2=2ax^6+(3a+8)x^4+(3a+8)x^2+2a$
- $y^2=x^6+2x^3+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$| The isogeny class factors as 1.169.aba 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.
| Subfield | Primitive Model |
| $\F_{13}$ | 2.13.a_aba |