Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 20 x + 177 x^{2} - 820 x^{3} + 1681 x^{4}$ |
Frobenius angles: | $\pm0.0953421678685$, $\pm0.292668588087$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.37025.1 |
Galois group: | $D_{4}$ |
Jacobians: | $14$ |
Isomorphism classes: | 14 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1019$ | $2750281$ | $4761110384$ | $7989635062025$ | $13422763808553979$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $22$ | $1636$ | $69082$ | $2827428$ | $115857102$ | $4750035886$ | $194753844622$ | $7984926680388$ | $327381981833482$ | $13422659764981956$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=15x^6+17x^5+20x^4+38x^3+34x^2+23x+3$
- $y^2=27x^6+4x^5+3x^4+8x^3+29x^2+2x+24$
- $y^2=28x^6+39x^5+17x^4+9x^3+10x^2+8x+26$
- $y^2=6x^6+17x^5+37x^4+5x^3+23x^2+39x+10$
- $y^2=22x^6+32x^5+11x^4+34x^3+34x+38$
- $y^2=28x^6+15x^5+15x^4+22x^3+24x^2+15x+17$
- $y^2=30x^6+29x^5+29x^4+9x^2+14x+15$
- $y^2=11x^6+20x^5+8x^4+33x^3+40x^2+24x+14$
- $y^2=17x^6+2x^5+21x^4+30x^3+2x^2+10x+15$
- $y^2=11x^6+22x^5+16x^4+6x^2+24x+11$
- $y^2=22x^6+16x^5+32x^4+34x^3+24x^2+23x+32$
- $y^2=6x^6+18x^5+12x^4+26x^3+28x^2+21x+34$
- $y^2=6x^6+34x^5+38x^4+12x^3+39x^2+25x+32$
- $y^2=16x^6+40x^5+18x^4+11x^3+13x^2+36x+19$
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 4.0.37025.1. |
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.u_gv | $2$ | (not in LMFDB) |