Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 19 x + 165 x^{2} - 779 x^{3} + 1681 x^{4}$ |
Frobenius angles: | $\pm0.0989395465808$, $\pm0.321602247780$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1518005.3 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1049$ | $2774605$ | $4764483521$ | $7987779813845$ | $13421787842370304$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $23$ | $1651$ | $69131$ | $2826771$ | $115848678$ | $4750009723$ | $194754154163$ | $7984930992771$ | $327382002674471$ | $13422659722105326$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=33x^6+24x^5+19x^4+35x^3+29x^2+11x+13$
- $y^2=22x^6+36x^5+19x^4+25x^3+26x^2+39x+3$
- $y^2=11x^6+28x^5+12x^4+24x^3+27x^2+16x+4$
- $y^2=30x^6+40x^5+39x^4+40x^3+24x^2+39$
- $y^2=3x^6+35x^5+28x^4+13x^3+32x^2+11x+8$
- $y^2=34x^6+23x^5+16x^4+28x^3+34x^2+39x+21$
- $y^2=30x^6+34x^5+27x^4+14x^3+25x^2+34x+11$
- $y^2=30x^6+26x^5+36x^4+11x^3+39x^2+12x+33$
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 4.0.1518005.3. |
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.t_gj | $2$ | (not in LMFDB) |