Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 71 x^{2} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.0833879852921$, $\pm0.916612014708$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-11}, \sqrt{17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $15$ |
| Isomorphism classes: | 28 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1611$ | $2595321$ | $4750104384$ | $7975444790889$ | $13422659511019851$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1540$ | $68922$ | $2822404$ | $115856202$ | $4750104526$ | $194754273882$ | $7984930894084$ | $327381934393962$ | $13422659711887300$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):
- $y^2=30 x^6+28 x^5+28 x^4+25 x^3+8 x^2+40 x+34$
- $y^2=2 x^6+19 x^5+27 x^4+38 x^3+33 x^2+31 x+29$
- $y^2=3 x^6+2 x^5+9 x^4+29 x^3+27 x^2+14 x+29$
- $y^2=18 x^6+12 x^5+13 x^4+10 x^3+39 x^2+2 x+10$
- $y^2=13 x^6+35 x^5+21 x^4+32 x^3+19 x^2+38 x+32$
- $y^2=37 x^6+5 x^5+3 x^4+28 x^3+32 x^2+23 x+28$
- $y^2=8 x^6+29 x^5+37 x^4+14 x^3+31 x^2+29$
- $y^2=7 x^6+10 x^5+17 x^4+2 x^3+22 x^2+10$
- $y^2=21 x^6+31 x^5+21 x^4+28 x^3+3 x^2+6 x+29$
- $y^2=2 x^6+10 x^5+9 x^4+22 x^3+13 x^2+38 x+31$
- $y^2=24 x^6+16 x^5+34 x^4+10 x^3+32 x^2+32 x+32$
- $y^2=21 x^6+14 x^5+40 x^4+19 x^3+28 x^2+28 x+28$
- $y^2=33 x^6+8 x^5+29 x^4+35 x^3+10 x^2+3 x+13$
- $y^2=15 x^6+25 x^5+15 x^4+2 x^3+37 x^2+8 x+1$
- $y^2=8 x^6+27 x^5+8 x^4+12 x^3+17 x^2+7 x+6$
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 \(\Q(\sqrt{-11}, \sqrt{17})\). |
| The base change of $A$ to $\F_{41^{2}}$ is 1.1681.act 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$ |
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 |
|---|---|---|
| 2.41.a_ct | $4$ | (not in LMFDB) |