Invariants
Base field: | $\F_{7^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 11 x + 49 x^{2} )^{2}$ |
$1 - 22 x + 219 x^{2} - 1078 x^{3} + 2401 x^{4}$ | |
Frobenius angles: | $\pm0.212295615010$, $\pm0.212295615010$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $8$ |
This isogeny class is not simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1521$ | $5659641$ | $13908900096$ | $33282226355625$ | $79810905102881121$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $28$ | $2356$ | $118222$ | $5773348$ | $282541228$ | $13841594206$ | $678223216972$ | $33232917111748$ | $1628413442812078$ | $79792265250964756$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+x^5+ax^4+6x^3+(6a+1)x^2+(6a+4)x+3$
- $y^2=(4a+5)x^6+(5a+6)x^5+ax^4+(2a+1)x^3+(4a+6)x^2+(5a+3)x+a+4$
- $y^2=x^6+5x^3+6$
- $y^2=(6a+5)x^6+(4a+2)x^5+(a+2)x^4+4x^3+(5a+4)x^2+5x+4$
- $y^2=(6a+3)x^6+(2a+3)x^5+(a+3)x^4+(5a+3)x^3+(4a+1)x^2+(2a+5)x+4$
- $y^2=4x^6+(3a+4)x^5+(2a+4)x^4+(6a+4)x^3+(2a+6)x^2+(6a+2)x+6a+3$
- $y^2=(5a+5)x^6+5ax^5+2ax^4+(2a+6)x^3+(2a+5)x^2+(a+1)x+3a+4$
- $y^2=(2a+2)x^6+5ax^5+2ax^4+(4a+1)x^3+(6a+2)x^2+(6a+6)x+2a+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$The isogeny class factors as 1.49.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.
Subfield | Primitive Model |
$\F_{7}$ | 2.7.ak_bn |
$\F_{7}$ | 2.7.a_al |
$\F_{7}$ | 2.7.k_bn |