Invariants
| Base field: | $\F_{43}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 43 x^{2} )^{2}$ |
| $1 - 86 x^{2} + 1849 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $1$, $1$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{43}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $6$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3, 7$ |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1764$ | $3111696$ | $6321204036$ | $11662935330816$ | $21611482019267364$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $44$ | $1678$ | $79508$ | $3411406$ | $147008444$ | $6321045022$ | $271818611108$ | $11688186602398$ | $502592611936844$ | $21611481725250478$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=x^6+20 x^5+16 x^4+19 x^3+12 x^2+12 x+34$
- $y^2=28 x^6+20 x^5+25 x^4+28 x^3+12 x^2+39 x+4$
- $y^2=x^6+28 x^3+22$
- $y^2=x^6+x^3+27$
- $y^2=x^6+x^3+22$
- $y^2=14 x^6+23 x^5+9 x^4+41 x^2+27 x+13$
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 the quaternion algebra over \(\Q(\sqrt{43}) \) ramified at both real infinite places. |
| The base change of $A$ to $\F_{43^{2}}$ is 1.1849.adi 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $43$ and $\infty$. |
Base change
This is a primitive isogeny class.