Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 20 x^{2} - 66 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.0987454850593$, $\pm0.515199656070$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.348480.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $70$ | $14980$ | $1702750$ | $209779920$ | $25965766750$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $126$ | $1278$ | $14326$ | $161226$ | $1774878$ | $19489266$ | $214354846$ | $2358079398$ | $25938032526$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+x^5+7x^4+10x^2+8x+10$
- $y^2=2x^6+4x^5+3x^4+3x^3+4x^2+4x+10$
- $y^2=5x^6+3x^5+x^3+10x^2+3x+2$
- $y^2=9x^6+9x^5+7x^4+7x^3+5x^2+4x+6$
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 4.0.348480.1. |
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.g_u | $2$ | 2.121.e_afu |