Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 11 x^{2} )( 1 - 2 x + 11 x^{2} )$ |
$1 - 6 x + 30 x^{2} - 66 x^{3} + 121 x^{4}$ | |
Frobenius angles: | $\pm0.293962833700$, $\pm0.402508885479$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 16 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $80$ | $17920$ | $1946000$ | $216186880$ | $25820762000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $146$ | $1458$ | $14766$ | $160326$ | $1768898$ | $19485906$ | $214368286$ | $2357954118$ | $25937419826$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=9x^6+6x^5+2x^4+6x^3+2x^2+6x+9$
- $y^2=8x^6+6x^5+3x^4+4x^3+9x^2+10x+7$
- $y^2=10x^6+10x^5+4x^4+6x^3+4x^2+10x+10$
- $y^2=2x^6+10x^5+9x^4+9x^3+9x^2+10x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$The isogeny class factors as 1.11.ae $\times$ 1.11.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ac_o | $2$ | 2.121.y_nm |
2.11.c_o | $2$ | 2.121.y_nm |
2.11.g_be | $2$ | 2.121.y_nm |