Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x^{2} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.220721427874$, $\pm0.779278572126$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-67}, \sqrt{97})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $21$ |
| Isomorphism classes: | 28 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1667$ | $2778889$ | $4750176512$ | $8002669552201$ | $13422659125827827$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1652$ | $68922$ | $2832036$ | $115856202$ | $4750248782$ | $194754273882$ | $7984916850628$ | $327381934393962$ | $13422658941503252$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):
- $y^2=7 x^6+29 x^5+6 x^4+36 x^3+17 x^2+25 x+34$
- $y^2=x^6+10 x^5+36 x^4+11 x^3+20 x^2+27 x+40$
- $y^2=19 x^6+25 x^5+31 x^4+26 x^3+20 x^2+20 x+31$
- $y^2=33 x^6+5 x^5+16 x^4+x^3+4 x^2+40 x+16$
- $y^2=3 x^6+8 x^5+22 x^4+30 x^3+32 x^2+30 x+21$
- $y^2=18 x^6+7 x^5+9 x^4+16 x^3+28 x^2+16 x+3$
- $y^2=5 x^6+3 x^5+14 x^4+30 x^3+4 x^2+36 x+21$
- $y^2=4 x^6+17 x^5+17 x^4+x^3+4 x^2+5 x+38$
- $y^2=24 x^6+20 x^5+20 x^4+6 x^3+24 x^2+30 x+23$
- $y^2=25 x^6+38 x^5+36 x^4+31 x^3+x^2+10 x+30$
- $y^2=23 x^6+10 x^5+36 x^4+16 x^3+12 x^2+23 x+24$
- $y^2=15 x^6+19 x^5+11 x^4+14 x^3+31 x^2+15 x+21$
- $y^2=26 x^6+30 x^5+20 x^4+35 x^3+23 x^2+23 x+40$
- $y^2=2 x^6+x^5+35 x^4+19 x^2+14 x+19$
- $y^2=12 x^6+6 x^5+5 x^4+32 x^2+2 x+32$
- $y^2=31 x^6+17 x^5+9 x^4+9 x^3+x^2+15 x+28$
- $y^2=22 x^6+20 x^5+13 x^4+13 x^3+6 x^2+8 x+4$
- $y^2=19 x^6+10 x^5+39 x^4+12 x^3+31 x^2+32 x+13$
- $y^2=32 x^6+19 x^5+29 x^4+31 x^3+22 x^2+28 x+37$
- $y^2=x^6+6 x^5+38 x^4+40 x^3+39 x^2+10 x+1$
- $y^2=29 x^6+35 x^5+28 x^4+4 x^3+30 x^2+28 x+1$
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 \(\Q(\sqrt{-67}, \sqrt{97})\). |
| The base change of $A$ to $\F_{41^{2}}$ is 1.1681.ap 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6499}) \)$)$ |
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_p | $4$ | (not in LMFDB) |