Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 98 x^{2} - 574 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.0312551325813$, $\pm0.468744867419$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{33})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 30 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1192$ | $2822656$ | $4726042792$ | $7967386894336$ | $13419284379958312$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $1682$ | $68572$ | $2819550$ | $115827068$ | $4750104242$ | $194754151676$ | $7984917237694$ | $327381883464412$ | $13422659310152402$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):
- $y^2=3 x^6+18 x^5+38 x^4+15 x^3+33 x^2+8 x+34$
- $y^2=19 x^6+2 x^5+10 x^4+34 x^3+2 x^2+3 x+11$
- $y^2=28 x^6+10 x^5+24 x^4+8 x^3+33 x^2+14 x+3$
- $y^2=38 x^6+36 x^5+24 x^4+12 x^3+40 x^2+26 x+28$
- $y^2=15 x^6+20 x^5+33 x^4+36 x^3+39 x^2+5 x+30$
- $y^2=22 x^6+37 x^5+10 x^4+23 x^3+31 x^2+24 x+28$
- $y^2=11 x^6+11 x^5+6 x^4+40 x^3+33 x^2+33 x+17$
- $y^2=35 x^6+16 x^5+16 x^4+31 x^3+19 x^2+x+11$
- $y^2=16 x^6+22 x^5+3 x^4+25 x^3+5 x^2+35 x+29$
- $y^2=40 x^6+10 x^5+37 x^4+31 x^3+6 x^2+39 x+27$
- $y^2=18 x^6+7 x^5+26 x^4+38 x^3+18 x^2+23 x+19$
- $y^2=36 x^6+27 x^5+33 x^4+33 x^2+14 x+36$
- $y^2=30 x^6+5 x^4+6 x^3+38 x^2+28 x+2$
- $y^2=19 x^6+7 x^5+36 x^4+36 x^2+34 x+19$
- $y^2=19 x^6+5 x^5+22 x^4+3 x^3+26 x^2+25 x+7$
- $y^2=27 x^6+25 x^5+35 x^4+33 x^3+18 x^2+34 x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41^{4}}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{33})\). |
| The base change of $A$ to $\F_{41^{4}}$ is 1.2825761.aepm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-33}) \)$)$ |
- Endomorphism algebra over $\F_{41^{2}}$
The base change of $A$ to $\F_{41^{2}}$ is the simple isogeny class 2.1681.a_aepm and its endomorphism algebra is \(\Q(i, \sqrt{33})\).
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.o_du | $2$ | (not in LMFDB) |
| 2.41.a_aq | $8$ | (not in LMFDB) |
| 2.41.a_q | $8$ | (not in LMFDB) |