Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 26 x^{2} - 66 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.210408938282$, $\pm0.463259653430$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.18000.1 |
Galois group: | $C_4$ |
Jacobians: | $12$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $76$ | $16720$ | $1846876$ | $214283520$ | $25994412316$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $138$ | $1386$ | $14638$ | $161406$ | $1775418$ | $19494306$ | $214325278$ | $2357772246$ | $25937249898$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+9x^5+3x^3+6x+10$
- $y^2=8x^6+x^5+8x^4+4x^3+3x^2+2x+2$
- $y^2=4x^6+6x^5+10x^4+6x^3+4x^2+7x+8$
- $y^2=2x^6+5x^5+7x^4+8x^3+8x$
- $y^2=5x^6+x^5+3x^4+7x^3+7x^2+x$
- $y^2=6x^6+6x^5+8x^4+7x^3+3x^2+7x+2$
- $y^2=8x^5+8x^4+6x^3+2x^2+8x$
- $y^2=7x^6+10x^5+x^4+5x^2+6x+10$
- $y^2=7x^6+3x^5+9x^4+7x^3+9x^2+8x+8$
- $y^2=2x^6+5x^3+7x+7$
- $y^2=10x^6+10x^5+5x^4+3x^3+4x^2+7x$
- $y^2=7x^5+10x^4+6x^3+8x^2+2x+7$
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 4.0.18000.1. |
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_ba | $2$ | 2.121.q_ew |