Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 12 x + 41 x^{2} )( 1 - 7 x + 41 x^{2} )$ |
$1 - 19 x + 166 x^{2} - 779 x^{3} + 1681 x^{4}$ | |
Frobenius angles: | $\pm0.113551764296$, $\pm0.315918729109$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $10$ |
Isomorphism classes: | 52 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1050$ | $2778300$ | $4768444800$ | $7989990724800$ | $13422569265656250$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $23$ | $1653$ | $69188$ | $2827553$ | $115855423$ | $4750048098$ | $194754284503$ | $7984931133313$ | $327382002383828$ | $13422659724515973$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=35x^6+22x^5+32x^4+36x^3+23x^2+22x+29$
- $y^2=36x^6+37x^5+5x^4+36x^3+24x^2+22x+3$
- $y^2=12x^6+25x^5+5x^4+18x^3+2x^2+5x+5$
- $y^2=15x^6+34x^5+34x^4+2x^3+14x^2+38x+40$
- $y^2=6x^6+2x^5+31x^4+40x^3+25x^2+3x+11$
- $y^2=7x^6+9x^5+20x^4+35x^3+16x^2+34x+27$
- $y^2=22x^6+27x^5+14x^4+31x^3+33x^2+18x+34$
- $y^2=14x^6+13x^5+37x^4+38x^3+30x^2+14x+18$
- $y^2=22x^6+30x^5+16x^4+13x^3+7x^2+39x+7$
- $y^2=9x^6+31x^5+3x^4+39x^3+12x^2+19x+6$
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 $\times$ 1.41.ah 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.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.41.af_ac | $2$ | (not in LMFDB) |
2.41.f_ac | $2$ | (not in LMFDB) |
2.41.t_gk | $2$ | (not in LMFDB) |