Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 82 x^{2} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.0487319457575$, $\pm0.951268054243$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{42})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1768$ | $3125824$ | $6321266536$ | $11667525682176$ | $21611482301574568$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $44$ | $1686$ | $79508$ | $3412750$ | $147008444$ | $6321170022$ | $271818611108$ | $11688195639454$ | $502592611936844$ | $21611482289864886$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=11 x^6+41 x^5+42 x^4+33 x^2+17 x+5$
- $y^2=7 x^6+4 x^5+11 x^4+21 x^2+27 x+30$
- $y^2=7 x^6+13 x^5+4 x^4+41 x^3+14 x^2+25 x+25$
- $y^2=39 x^6+41 x^5+35 x^4+13 x^3+37 x^2+16 x+24$
- $y^2=21 x^6+6 x^5+19 x^4+18 x^2+2 x+16$
- $y^2=35 x^6+15 x^5+8 x^4+2 x^3+37 x^2+11 x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{43^{2}}$.
Endomorphism algebra over $\F_{43}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{42})\). |
| The base change of $A$ to $\F_{43^{2}}$ is 1.1849.ade 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$ |
Base change
This is a primitive isogeny class.