Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 19 x + 166 x^{2} - 817 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.0801255036674$, $\pm0.340545434875$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-206 -22 \sqrt{41}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1180$ | $3365360$ | $6333376240$ | $11686751057600$ | $21608267787046900$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $25$ | $1821$ | $79660$ | $3418377$ | $146986575$ | $6321195894$ | $271818437125$ | $11688207638673$ | $502592684547700$ | $21611482637880061$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=39 x^6+4 x^5+8 x^4+2 x^3+32 x+11$
- $y^2=3 x^6+40 x^5+12 x^4+25 x^3+9 x^2+35 x+16$
- $y^2=31 x^6+5 x^5+9 x^4+x^3+8 x^2+15 x+3$
- $y^2=8 x^6+36 x^5+5 x^4+5 x^3+28 x^2+41 x+5$
- $y^2=16 x^6+32 x^5+21 x^4+23 x^3+26 x^2+34 x+3$
- $y^2=16 x^6+28 x^5+23 x^4+5 x^3+23 x^2+8 x+22$
- $y^2=2 x^6+37 x^5+29 x^4+42 x^3+34 x^2+16 x+29$
- $y^2=14 x^6+8 x^5+12 x^4+21 x^3+28 x^2+39 x+33$
- $y^2=31 x^6+40 x^5+31 x^4+39 x^3+24 x^2+5 x+18$
- $y^2=22 x^6+30 x^5+6 x^4+25 x^3+27 x^2+15 x$
- $y^2=33 x^6+6 x^5+33 x^4+18 x^3+39 x^2+3 x+4$
- $y^2=22 x^6+17 x^5+25 x^4+38 x^3+31 x^2+22 x+42$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{43}$.
Endomorphism algebra over $\F_{43}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-206 -22 \sqrt{41}})\). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.43.t_gk | $2$ | (not in LMFDB) |