Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x + 13 x^{2} )( 1 + 2 x + 13 x^{2} )$ |
| $1 - x + 20 x^{2} - 13 x^{3} + 169 x^{4}$ | |
| Frobenius angles: | $\pm0.363422825076$, $\pm0.589456187511$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $176$ | $35904$ | $4868864$ | $813010176$ | $137924466416$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $209$ | $2218$ | $28465$ | $371473$ | $4822598$ | $62736085$ | $815821249$ | $10604753554$ | $137857485689$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=5 x^6+3 x^5+11 x^3+12 x+2$
- $y^2=x^5+10 x^4+4 x^3+12 x^2+11 x+1$
- $y^2=6 x^6+x^5+x^3+4 x^2+3 x+3$
- $y^2=4 x^6+9 x^5+8 x^4+12 x^3+12 x^2+9 x+2$
- $y^2=7 x^6+6 x^5+2 x^4+8 x^3+12 x^2+x+3$
- $y^2=5 x^6+3 x^5+5 x^4+10 x^3+5 x^2+7 x+6$
- $y^2=x^6+7 x^5+3 x^4+12 x^3+11 x^2+12 x+7$
- $y^2=2 x^6+3 x^5+9 x^4+12 x^3+10 x^2+3 x+6$
- $y^2=7 x^6+5 x^5+9 x^4+11 x^3+8 x+12$
- $y^2=12 x^6+4 x^5+11 x^4+11 x^3+8 x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$| The isogeny class factors as 1.13.ad $\times$ 1.13.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.