Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 19 x + 165 x^{2} - 817 x^{3} + 1849 x^{4}$ |
| Frobenius angles: | $\pm0.0635826053657$, $\pm0.344751241674$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-33 +6 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1179$ | $3361329$ | $6328814229$ | $11684077082541$ | $21607248381873264$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $25$ | $1819$ | $79603$ | $3417595$ | $146979640$ | $6321149863$ | $271818151441$ | $11688205418659$ | $502592664546799$ | $21611482483233814$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=37 x^6+11 x^5+21 x^4+19 x^3+23 x^2+9 x+21$
- $y^2=20 x^6+7 x^5+39 x^4+17 x^3+18 x^2+18 x+37$
- $y^2=30 x^6+35 x^4+31 x^3+20 x^2+24 x+9$
- $y^2=19 x^6+37 x^5+15 x^4+4 x^3+19 x^2+17 x+2$
- $y^2=9 x^6+x^5+5 x^4+23 x^3+6 x^2+42 x+15$
- $y^2=3 x^6+16 x^5+30 x^4+23 x^3+33 x^2+10 x+27$
- $y^2=10 x^6+23 x^5+22 x^4+39 x^3+41 x^2+30 x+22$
- $y^2=12 x^6+34 x^5+16 x^4+18 x^3+20 x^2+39 x+8$
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 \(\Q(\sqrt{-33 +6 \sqrt{5}})\). |
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_gj | $2$ | (not in LMFDB) |