Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 41 x^{2} )^{2}$ |
| $1 - 82 x^{2} + 1681 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $1$, $1$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{41}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1600$ | $2560000$ | $4749966400$ | $7965941760000$ | $13422659078440000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1518$ | $68922$ | $2819038$ | $115856202$ | $4749828558$ | $194754273882$ | $7984913926078$ | $327381934393962$ | $13422658846727598$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=31 x^6+8 x^5+17 x^4+3 x^3+32 x^2+37 x+7$
- $y^2=38 x^6+x^5+8 x^4+x^2+16 x+6$
- $y^2=16 x^6+11 x^5+28 x^4+11 x^3+32 x^2+6 x+35$
- $y^2=34 x^6+11 x^5+9 x^4+11 x^2+22 x+33$
- $y^2=26 x^5+19 x^3+3 x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41^{2}}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{41}) \) ramified at both real infinite places. |
| The base change of $A$ to $\F_{41^{2}}$ is 1.1681.ade 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $41$ and $\infty$. |
Base change
This is a primitive isogeny class.