Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 13 x^{2} )^{2}$ |
$1 - 10 x + 51 x^{2} - 130 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.256122854178$, $\pm0.256122854178$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $81$ | $29241$ | $5143824$ | $835152201$ | $138435340761$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $172$ | $2338$ | $29236$ | $372844$ | $4825798$ | $62723308$ | $815617828$ | $10604262634$ | $137858775772$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+9x^3+1$
- $y^2=11x^6+2x^5+2x^4+7x^3+2x^2+2x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.