Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 17 x + 138 x^{2} - 697 x^{3} + 1681 x^{4}$ |
Frobenius angles: | $\pm0.0660990950969$, $\pm0.386534833598$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4241900.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1106$ | $2802604$ | $4752389096$ | $7977152658944$ | $13418955038131426$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $25$ | $1669$ | $68956$ | $2823009$ | $115824225$ | $4749986098$ | $194754702265$ | $7984931155329$ | $327381947429596$ | $13422659211124629$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=35x^6+34x^5+9x^4+18x^3+32x^2+28x+18$
- $y^2=24x^6+x^5+6x^4+4x^3+14x^2+36x+35$
- $y^2=9x^6+38x^5+15x^4+33x^2+24x+17$
- $y^2=24x^6+37x^5+3x^4+24x^3+13x^2+34x+10$
- $y^2=17x^6+31x^5+38x^4+23x^3+29x^2+11x+3$
- $y^2=28x^6+36x^5+34x^4+10x^3+29x+26$
- $y^2=7x^6+31x^5+11x^4+23x^3+13x^2+30x+1$
- $y^2=24x^6+34x^5+21x^3+5x^2+9x+29$
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 4.0.4241900.1. |
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.r_fi | $2$ | (not in LMFDB) |