Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x - 46 x^{2} + 82 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.200112285954$, $\pm0.915515338997$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-34 +2 \sqrt{129}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1720$ | $2669440$ | $4786830520$ | $7987989544960$ | $13425755563123000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $44$ | $1586$ | $69452$ | $2826846$ | $115882924$ | $4750218578$ | $194754294604$ | $7984926433726$ | $327381878642732$ | $13422659283168626$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=21 x^6+8 x^5+24 x^4+19 x^3+3 x^2+30 x+24$
- $y^2=23 x^6+13 x^5+24 x^4+3 x^3+12 x^2+38$
- $y^2=11 x^6+23 x^5+6 x^4+2 x^3+32 x^2+20 x+24$
- $y^2=20 x^6+6 x^5+4 x^4+12 x^3+5 x^2+24 x+13$
- $y^2=5 x^6+17 x^5+37 x^4+3 x^3+14 x^2+35 x+32$
- $y^2=9 x^6+6 x^5+24 x^4+21 x^3+32 x^2+20 x+20$
- $y^2=18 x^6+17 x^5+16 x^4+13 x^3+5 x^2+20 x+2$
- $y^2=x^6+38 x^5+6 x^4+24 x^3+25 x^2+37$
- $y^2=12 x^6+31 x^5+18 x^4+27 x^3+34 x^2+15 x+2$
- $y^2=20 x^6+33 x^5+40 x^4+36 x^3+12 x^2+29 x+20$
- $y^2=35 x^5+20 x^4+25 x^3+40 x^2+8 x$
- $y^2=15 x^6+36 x^5+9 x^4+39 x^3+13 x^2+11 x+20$
- $y^2=32 x^6+3 x^5+22 x^4+24 x^3+30 x^2+22 x+28$
- $y^2=14 x^6+24 x^5+19 x^4+33 x^3+23 x^2+9 x+19$
- $y^2=15 x^6+36 x^5+12 x^4+25 x^3+14 x^2+39 x+25$
- $y^2=4 x^6+29 x^5+27 x^4+34 x^3+28 x^2+17 x+25$
- $y^2=21 x^6+2 x^5+5 x^4+37 x^3+16 x^2+21$
- $y^2=10 x^6+4 x^5+25 x^4+32 x^3+4 x^2+22 x+28$
- $y^2=20 x^6+7 x^5+35 x^4+19 x^3+16 x^2+27 x+12$
- $y^2=10 x^6+22 x^5+5 x^4+8 x^3+23 x^2+5 x+30$
- $y^2=31 x^6+32 x^5+5 x^4+31 x^3+22 x^2+31 x+8$
- $y^2=38 x^6+26 x^5+40 x^4+14 x^3+22 x^2+14 x+29$
- $y^2=13 x^6+10 x^5+25 x^4+2 x^3+16 x^2+20 x+8$
- $y^2=39 x^6+3 x^5+18 x^4+15 x^3+23 x^2+30 x+5$
- $y^2=13 x^6+2 x^5+4 x^4+36 x^3+20 x$
- $y^2=34 x^6+9 x^5+22 x^4+25 x^3+31 x^2+17 x+13$
- $y^2=30 x^6+6 x^5+29 x^4+39 x^3+2 x^2+17 x+28$
- $y^2=25 x^6+38 x^5+35 x^4+7 x^3+21 x^2+14 x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-34 +2 \sqrt{129}})\). |
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.ac_abu | $2$ | (not in LMFDB) |