Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 20 x + 180 x^{2} - 820 x^{3} + 1681 x^{4}$ |
Frobenius angles: | $\pm0.149793866772$, $\pm0.266106536017$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.194816.2 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1022$ | $2761444$ | $4773640382$ | $7997152869776$ | $13425683554533502$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $22$ | $1642$ | $69262$ | $2830086$ | $115882302$ | $4750193962$ | $194754344422$ | $7984924623678$ | $327381939284182$ | $13422659423960202$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+10x^5+36x^4+22x^3+7x^2+15x+34$
- $y^2=17x^6+5x^5+26x^4+15x^3+23x^2+3x+19$
- $y^2=38x^6+33x^5+4x^4+5x^3+4x+27$
- $y^2=27x^6+17x^5+8x^4+13x^3+6x^2+38x+7$
- $y^2=11x^6+28x^5+36x^4+28x^3+2x^2+22x+13$
- $y^2=11x^6+24x^5+30x^3+31x^2+27x+24$
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 4.0.194816.2. |
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_gy | $2$ | (not in LMFDB) |