Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 2 x + 13 x^{2} )^{2}$ |
| $1 + 4 x + 30 x^{2} + 52 x^{3} + 169 x^{4}$ | |
| Frobenius angles: | $\pm0.589456187511$, $\pm0.589456187511$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $256$ | $36864$ | $4528384$ | $807469056$ | $138753270016$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $214$ | $2058$ | $28270$ | $373698$ | $4825798$ | $62719290$ | $815802334$ | $10604736114$ | $137857087414$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=3 x^6+3 x^5+11 x^4+4 x^3+11 x^2+3 x+3$
- $y^2=3 x^6+8 x^5+9 x^4+4 x^3+12 x^2+7 x+10$
- $y^2=2 x^6+11$
- $y^2=8 x^6+2 x^4+2 x^2+8$
- $y^2=12 x^6+10 x^5+6 x^4+5 x^3+6 x^2+10 x+12$
- $y^2=10 x^6+10 x^4+10 x^2+10$
- $y^2=4 x^6+7 x^5+10 x^4+5 x^3+10 x^2+7 x+4$
- $y^2=2 x^6+6 x^3+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$| The isogeny class factors as 1.13.c 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.