Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 19 x + 167 x^{2} - 779 x^{3} + 1681 x^{4}$ |
Frobenius angles: | $\pm0.127582089543$, $\pm0.309683357545$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1233477.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1051$ | $2781997$ | $4772407075$ | $7992190531525$ | $13423328697807856$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $23$ | $1655$ | $69245$ | $2828331$ | $115861978$ | $4750081967$ | $194754344353$ | $7984930469011$ | $327381995121515$ | $13422659681255630$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=23x^6+3x^5+38x^4+27x^3+29x^2+8x+6$
- $y^2=39x^6+14x^5+26x^4+25x^3+6x^2+22x+23$
- $y^2=7x^6+32x^5+24x^4+21x^3+6x^2+26x+13$
- $y^2=27x^6+34x^5+15x^4+35x^3+15x^2+40x+28$
- $y^2=22x^6+x^5+29x^4+6x^3+x^2+11x+14$
- $y^2=4x^6+37x^5+16x^4+32x^3+22x+40$
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 4.0.1233477.1. |
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.t_gl | $2$ | (not in LMFDB) |