Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 12 x + 41 x^{2} )( 1 - 8 x + 41 x^{2} )$ |
$1 - 20 x + 178 x^{2} - 820 x^{3} + 1681 x^{4}$ | |
Frobenius angles: | $\pm0.113551764296$, $\pm0.285223287477$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $20$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1020$ | $2754000$ | $4765285980$ | $7992152064000$ | $13423760198305500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $22$ | $1638$ | $69142$ | $2828318$ | $115865702$ | $4750093638$ | $194754096902$ | $7984927070398$ | $327381978057142$ | $13422659728370598$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=38x^5+23x^4+37x^3+25x^2+28x$
- $y^2=13x^6+2x^5+9x^4+40x^3+24x^2+8x+38$
- $y^2=20x^6+32x^5+12x^4+10x^3+9x^2+27x+31$
- $y^2=24x^6+31x^5+14x^4+5x^3+13x^2+x+13$
- $y^2=37x^6+25x^5+15x^4+35x^3+15x^2+25x+37$
- $y^2=35x^6+22x^5+16x^4+2x^3+22x^2+30x+11$
- $y^2=12x^6+31x^5+12x^4+36x^3+33x^2+17x+30$
- $y^2=38x^6+3x^5+3x^4+29x^3+19x^2+22x+3$
- $y^2=5x^6+15x^5+40x^4+30x^3+40x^2+15x+5$
- $y^2=15x^6+18x^5+6x^4+20x^3+6x^2+18x+15$
- $y^2=34x^6+11x^5+39x^4+9x^3+10x^2+29x+14$
- $y^2=35x^6+11x^5+40x^4+29x^3+37x^2+26x+14$
- $y^2=12x^6+12x^5+9x^4+31x^3+9x^2+12x+12$
- $y^2=27x^6+8x^5+35x^4+4x^3+12x^2+27x+34$
- $y^2=12x^6+25x^5+38x^4+3x^3+6x^2+34x+22$
- $y^2=20x^6+33x^4+29x^3+39x^2+39x+5$
- $y^2=34x^6+33x^5+38x^4+39x^3+38x^2+33x+34$
- $y^2=27x^6+16x^4+28x^3+16x^2+27$
- $y^2=15x^6+32x^5+31x^4+31x^3+31x^2+32x+15$
- $y^2=38x^6+24x^5+9x^4+38x^3+40x^2+11x+7$
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.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.