# Properties

 Label 2.11.ai_bg Base Field $\F_{11}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## 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

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

• $y^2=2x^6+2x^5+9x^4+9x^3+10x^2+6x+8$
• $y^2=6x^6+8x^5+3x^4+x^3+3x+10$
• $y^2=10x^6+3x^5+2x^4+2x^2+8x+10$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 58 14500 1760938 210250000 25769191258 3138431750500 379935240802378 45953639424000000 5559842595347832058 672749994932236862500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 122 1324 14358 160004 1771562 19496684 214377118 2357916004 25937424602

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{6})$$.
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{4}}$ is 1.14641.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Remainder of endomorphism lattice by field
• 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.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.i_bg $2$ 2.121.a_afm 2.11.i_bg $4$ (not in LMFDB) 2.11.a_ak $8$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.i_bg $2$ 2.121.a_afm 2.11.i_bg $4$ (not in LMFDB) 2.11.a_ak $8$ (not in LMFDB) 2.11.a_k $8$ (not in LMFDB) 2.11.ag_x $24$ (not in LMFDB) 2.11.g_x $24$ (not in LMFDB)