Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 13 x^{2} )^{2}$ |
$1 - 8 x + 42 x^{2} - 104 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.312832958189$, $\pm0.312832958189$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $5$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $100$ | $32400$ | $5244100$ | $829440000$ | $137678102500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $190$ | $2382$ | $29038$ | $370806$ | $4818670$ | $62722302$ | $815731678$ | $10604844006$ | $137859857950$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+2x^3+11$
- $y^2=x^6+5x^5+6x^4+9x^3+6x^2+5x+1$
- $y^2=2x^6+8x^3+11$
- $y^2=5x^6+8x^5+6x^4+12x^3+6x^2+8x+5$
- $y^2=7x^6+12x^4+12x^2+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.