Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 71 x^{2} - 369 x^{3} + 1681 x^{4}$ |
Frobenius angles: | $\pm0.211163249537$, $\pm0.527129909542$ |
Angle rank: | $2$ (numerical) |
Number field: | \(\Q(\zeta_{5})\) |
Galois group: | $C_4$ |
Jacobians: | $69$ |
Isomorphism classes: | 91 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1375$ | $2930125$ | $4755796375$ | $7984359145125$ | $13426146538750000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $33$ | $1743$ | $69003$ | $2825563$ | $115886298$ | $4750316583$ | $194753838543$ | $7984917674803$ | $327381925076613$ | $13422659245067598$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 69 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=36x^5+40$
- $y^2=27x^6+14x^5+25x^4+30x^3+9x^2+30x+34$
- $y^2=6x^6+15x^5+37x^4+6x^3+30x^2+7x+24$
- $y^2=40x^6+24x^5+11x^4+33x^3+2x^2+19x+13$
- $y^2=17x^6+12x^5+5x^4+14x^3+29x^2+25x+19$
- $y^2=35x^6+8x^5+26x^4+8x^3+16x^2+27x+17$
- $y^2=32x^6+32x^5+12x^4+2x^2+30x+4$
- $y^2=26x^6+36x^5+15x^4+13x^3+17x+22$
- $y^2=22x^6+2x^5+28x^4+32x^3+3x^2+37x+7$
- $y^2=26x^5+14x^4+22x^2+26x+22$
- $y^2=18x^6+34x^5+16x^4+26x^3+37x^2+26x+23$
- $y^2=35x^6+17x^5+31x^4+20x^3+22x^2+15x+29$
- $y^2=40x^6+16x^5+40x^4+33x^3+11x^2+2x+37$
- $y^2=15x^6+8x^5+27x^4+34x^3+39x^2+19x+38$
- $y^2=29x^6+39x^5+33x^4+15x^3+38x^2+32x+28$
- $y^2=29x^6+11x^5+12x^4+35x^3+33x^2+10x+29$
- $y^2=9x^6+19x^5+13x^4+39x^3+4x^2+36x+9$
- $y^2=34x^6+17x^5+36x^4+19x^3+7x^2+40x+30$
- $y^2=22x^6+19x^5+23x^4+36x^3+17x^2+6x+33$
- $y^2=15x^6+10x^5+20x^4+22x^3+20x^2+26x+13$
- and 49 more
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 \(\Q(\zeta_{5})\). |
Base change
This is a primitive isogeny class.