Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 30 x^{2} - 77 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.183470593443$, $\pm0.430420419745$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.39593.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $68$ | $16048$ | $1849328$ | $214465472$ | $25947340108$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $133$ | $1388$ | $14649$ | $161115$ | $1774774$ | $19501529$ | $214367025$ | $2357813684$ | $25936959093$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+10x^5+3x^4+3x^3+8x^2+5x+8$
- $y^2=x^6+4x^4+x^3+10x^2+10x+2$
- $y^2=6x^6+8x^5+x^4+5x^3+8x+2$
- $y^2=10x^6+6x^5+4x^4+5x^3+x^2+8x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$The endomorphism algebra of this simple isogeny class is 4.0.39593.1. |
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 |
---|---|---|
2.11.h_be | $2$ | 2.121.l_cm |