Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 35 x^{2} - 88 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.167863583547$, $\pm0.388927483238$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.62352.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $61$ | $15433$ | $1859524$ | $215583577$ | $25923018781$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $128$ | $1396$ | $14724$ | $160964$ | $1772894$ | $19500380$ | $214400260$ | $2357944300$ | $25936950368$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+6x^5+2x^4+4x^3+9x^2+2x+7$
- $y^2=7x^6+8x^5+2x^3+10x^2+10x+9$
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 4.0.62352.1. |
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.i_bj | $2$ | 2.121.g_ch |