Invariants
Base field: | $\F_{11}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 23 x^{2} - 66 x^{3} + 121 x^{4}$ |
Frobenius angles: | $\pm0.158432477193$, $\pm0.491765810526$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
Galois group: | $C_2^2$ |
Jacobians: | $9$ |
Isomorphism classes: | 9 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $73$ | $15841$ | $1774192$ | $212285241$ | $26023682473$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $132$ | $1332$ | $14500$ | $161586$ | $1776822$ | $19497078$ | $214349764$ | $2357947692$ | $25937651652$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+9x^5+5x^4+4x^3+7x^2+8$
- $y^2=8x^6+9x^5+x^4+7x^3+2x^2+8x+8$
- $y^2=10x^6+4x^5+10x^4+3x^3+x^2+6x+8$
- $y^2=2x^6+9x^5+x^4+10x^3+8x^2+3x+2$
- $y^2=2x^6+4x^5+8x^4+5x^3+10x^2+10$
- $y^2=10x^6+3x^5+5x^4+5x^3+4x^2+6x+2$
- $y^2=10x^6+3x^5+x^4+10x^3+4x^2+2x+10$
- $y^2=2x^6+6x^5+5x^4+5x^3+2x^2+8x+2$
- $y^2=3x^6+7x^5+3x^4+x^3+2x^2+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{6}}$.
Endomorphism algebra over $\F_{11}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\). |
The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.dxe 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.k_av and its endomorphism algebra is \(\Q(\sqrt{2}, \sqrt{-3})\). - Endomorphism algebra over $\F_{11^{3}}$
The base change of $A$ to $\F_{11^{3}}$ is the simple isogeny class 2.1331.a_dxe and its endomorphism algebra is \(\Q(\sqrt{2}, \sqrt{-3})\).
Base change
This is a primitive isogeny class.