Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 32 x^{2} - 88 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.0750991438595$, $\pm0.424900856141$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(i, \sqrt{6})\) |
Galois group: | $C_2^2$ |
Jacobians: | $3$ |
Isomorphism classes: | 4 |
This isogeny class is simple but not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $58$ | $14500$ | $1760938$ | $210250000$ | $25769191258$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $122$ | $1324$ | $14358$ | $160004$ | $1771562$ | $19496684$ | $214377118$ | $2357916004$ | $25937424602$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+2x^5+9x^4+9x^3+10x^2+6x+8$
- $y^2=6x^6+8x^5+3x^4+x^3+3x+10$
- $y^2=10x^6+3x^5+2x^4+2x^2+8x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Endomorphism algebra over $\F_{11}$The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\). |
The base change of $A$ to $\F_{11^{4}}$ is 1.14641.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$ |
- Endomorphism algebra over $\F_{11^{2}}$
The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.a_afm and its endomorphism algebra is \(\Q(i, \sqrt{6})\).
Base change
This is a primitive isogeny class.