Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x - 39 x^{2} - 86 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.117934720909$, $\pm0.784601387576$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{14})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 82 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1723$ | $3271977$ | $6281830564$ | $11698551310329$ | $21609012412706923$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1768$ | $79008$ | $3421828$ | $146991642$ | $6321556078$ | $271819526622$ | $11688202596676$ | $502592699947344$ | $21611482301574568$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=8 x^6+15 x^5+22 x^4+27 x^2+26 x+15$
- $y^2=20 x^6+8 x^5+16 x^4+26 x^3+33 x^2+17 x+41$
- $y^2=20 x^6+28 x^5+2 x^4+38 x^3+35 x^2+24 x+17$
- $y^2=19 x^6+30 x^5+32 x^4+2 x^3+32 x^2+24 x+20$
- $y^2=3 x^6+3 x^3+22$
- $y^2=7 x^6+3 x^5+27 x^4+33 x^3+38 x^2+7 x+10$
- $y^2=22 x^6+11 x^5+16 x^4+26 x^3+21 x^2+26 x+21$
- $y^2=35 x^6+15 x^5+5 x^4+13 x^3+27 x^2+39 x+14$
- $y^2=3 x^6+3 x^3+32$
- $y^2=16 x^6+19 x^5+3 x^4+37 x^3+41 x^2+14 x+36$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{43^{3}}$.
Endomorphism algebra over $\F_{43}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{14})\). |
| The base change of $A$ to $\F_{43^{3}}$ is 1.79507.ajq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$ |
Base change
This is a primitive isogeny class.