Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 19 x^{2} - 55 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.185239138481$, $\pm0.526008203156$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.795389.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $81$ | $16281$ | $1767339$ | $213427629$ | $26115152976$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $135$ | $1327$ | $14579$ | $162152$ | $1776411$ | $19487657$ | $214337299$ | $2357964157$ | $25937356950$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+6x^5+10x^4+8x^3+x^2+5x+7$
- $y^2=7x^6+10x^5+5x^4+9x^3+6x+2$
- $y^2=6x^6+7x^5+7x^4+9x^3+8x^2+10x+7$
- $y^2=7x^6+9x^4+2x^3+4x^2+x+10$
- $y^2=6x^6+8x^5+9x^4+10x^2+7x$
- $y^2=5x^6+10x^5+3x^4+10x^3+9x^2+3x+1$
- $y^2=2x^6+3x^5+3x^4+4x^3+2x^2+5x+9$
- $y^2=7x^6+2x^5+9x^4+10x^2+2x+10$
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 4.0.795389.1. |
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.f_t | $2$ | 2.121.n_cb |