Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$ |
$1 - 10 x + 47 x^{2} - 130 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.363422825076$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $77$ | $27489$ | $4868864$ | $811722681$ | $137321294957$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $164$ | $2218$ | $28420$ | $369844$ | $4822598$ | $62752708$ | $815806084$ | $10604753554$ | $137858748164$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+9x^5+10x^4+8x^3+11x^2+5x+1$
- $y^2=2x^6+12x^5+4x^4+3x^3+3x^2+10x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ah $\times$ 1.13.ad 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.