Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 11 x^{2} )( 1 - 4 x + 11 x^{2} )$ |
$1 - 10 x + 46 x^{2} - 110 x^{3} + 121 x^{4}$ | |
Frobenius angles: | $\pm0.140218899004$, $\pm0.293962833700$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $48$ | $13824$ | $1839600$ | $218087424$ | $26026361328$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $114$ | $1382$ | $14894$ | $161602$ | $1771938$ | $19487302$ | $214372894$ | $2358056642$ | $25937838354$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+2x^4+10x^3+2x^2+2$
- $y^2=9x^5+5x^4+x^3+5x^2+9x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$The isogeny class factors as 1.11.ag $\times$ 1.11.ae 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_ac | $2$ | 2.121.ai_gc |
2.11.c_ac | $2$ | 2.121.ai_gc |
2.11.k_bu | $2$ | 2.121.ai_gc |