Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 2 x + 7 x^{2} )^{2}$ |
$1 - 4 x + 18 x^{2} - 28 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.376624142786$, $\pm0.376624142786$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $3$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $36$ | $3600$ | $142884$ | $5760000$ | $274432356$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $70$ | $412$ | $2398$ | $16324$ | $116710$ | $825052$ | $5774398$ | $40362244$ | $282425350$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+4x^5+5x^4+2x^3+5x^2+4x+5$
- $y^2=x^6+4x^3+1$
- $y^2=5x^6+4x^4+4x^2+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$ |
Base change
This is a primitive isogeny class.