Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 41 x^{2} )^{2}$ |
$1 - 18 x + 163 x^{2} - 738 x^{3} + 1681 x^{4}$ | |
Frobenius angles: | $\pm0.251940962052$, $\pm0.251940962052$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $15$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1089$ | $2832489$ | $4802490000$ | $8003936949129$ | $13426077749128209$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $24$ | $1684$ | $69678$ | $2832484$ | $115885704$ | $4750094158$ | $194752973544$ | $7984913939524$ | $327381886101438$ | $13422659338393204$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=29x^6+27x^5+25x^4+35x^3+9x^2+26x+27$
- $y^2=6x^6+18x^5+6x^4+33x^3+17x^2+x+26$
- $y^2=18x^6+25x^5+6x^4+25x^3+24x^2+31x+4$
- $y^2=20x^6+28x^5+29x^4+5x^3+14x^2+29x+10$
- $y^2=4x^6+39x^5+26x^4+36x^3+35x^2+21x+16$
- $y^2=39x^6+3x^5+10x^4+34x^3+9x^2+34x+37$
- $y^2=33x^6+21x^5+30x^4+23x^3+6x^2+32x+37$
- $y^2=39x^6+30x^5+35x^4+38x^3+30x^2+12x+4$
- $y^2=14x^6+27x^5+34x^4+3x^3+21x^2+12x+31$
- $y^2=30x^6+20x^5+33x^4+10x^3+20x^2+2x+13$
- $y^2=28x^6+36x^5+16x^4+37x^3+18x^2+2x+12$
- $y^2=38x^6+29x^5+x^4+24x^3+23x^2+11x+25$
- $y^2=18x^6+16x^5+34x^4+18x^3+20x^2+32x+32$
- $y^2=11x^6+21x^5+40x^4+26x^3+37x^2+8x+7$
- $y^2=15x^6+28x^5+35x^4+12x^3+17x^2+38x+17$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$The isogeny class factors as 1.41.aj 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-83}) \)$)$ |
Base change
This is a primitive isogeny class.