Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 11 x + 41 x^{2} )( 1 - 8 x + 41 x^{2} )$ |
$1 - 19 x + 170 x^{2} - 779 x^{3} + 1681 x^{4}$ | |
Frobenius angles: | $\pm0.171113726078$, $\pm0.285223287477$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $5$ |
Isomorphism classes: | 13 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1054$ | $2793100$ | $4784299936$ | $7998723366400$ | $13425475059245854$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $23$ | $1661$ | $69416$ | $2830641$ | $115880503$ | $4750156658$ | $194754108943$ | $7984923913441$ | $327381937417736$ | $13422659374671101$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+7x^5+39x^4+40x^3+3x^2+11x+11$
- $y^2=22x^6+25x^5+14x^4+15x^3+26x^2+19x+27$
- $y^2=11x^6+28x^5+31x^4+12x^3+9x^2+11x+35$
- $y^2=4x^6+19x^5+19x^4+28x^3+38x^2+17x+15$
- $y^2=18x^6+36x^5+36x^4+30x^3+10x^2+11x+29$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$The isogeny class factors as 1.41.al $\times$ 1.41.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.