Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 12 x + 41 x^{2} )^{2}$ |
$1 - 24 x + 226 x^{2} - 984 x^{3} + 1681 x^{4}$ | |
Frobenius angles: | $\pm0.113551764296$, $\pm0.113551764296$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $3$ |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $900$ | $2624400$ | $4715568900$ | $7982207078400$ | $13423713390562500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $18$ | $1558$ | $68418$ | $2824798$ | $115865298$ | $4750252918$ | $194755685058$ | $7984936067518$ | $327382006596498$ | $13422659732208598$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+17x^4+30x^3+35x^2+36$
- $y^2=29x^6+26x^4+26x^2+29$
- $y^2=27x^6+10x^5+9x^4+19x^3+x^2+31x+38$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$The isogeny class factors as 1.41.am 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
Base change
This is a primitive isogeny class.