# Properties

 Label 2.11.ag_x Base field $\F_{11}$ Dimension $2$ $p$-rank $2$ Ordinary Yes Supersingular No Simple Yes Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian Yes

## 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

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 9 curves, and hence is principally polarizable:

• $y^2=7x^6+9x^5+5x^4+4x^3+7x^2+8$
• $y^2=8x^6+9x^5+x^4+7x^3+2x^2+8x+8$
• $y^2=10x^6+4x^5+10x^4+3x^3+x^2+6x+8$
• $y^2=2x^6+9x^5+x^4+10x^3+8x^2+3x+2$
• $y^2=2x^6+4x^5+8x^4+5x^3+10x^2+10$
• $y^2=10x^6+3x^5+5x^4+5x^3+4x^2+6x+2$
• $y^2=10x^6+3x^5+x^4+10x^3+4x^2+2x+10$
• $y^2=2x^6+6x^5+5x^4+5x^3+2x^2+8x+2$
• $y^2=3x^6+7x^5+3x^4+x^3+2x^2+3$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 73 15841 1774192 212285241 26023682473 3147757252864 379942925705857 45947775208065129 5559917317706062192 672755884050430171201

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 132 1332 14500 161586 1776822 19497078 214349764 2357947692 25937651652

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.dxe 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$
All geometric endomorphisms are defined over $\F_{11^{6}}$.
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.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.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.g_x $2$ 2.121.k_av 2.11.a_ak $3$ (not in LMFDB) 2.11.g_x $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.g_x $2$ 2.121.k_av 2.11.a_ak $3$ (not in LMFDB) 2.11.g_x $3$ (not in LMFDB) 2.11.a_k $12$ (not in LMFDB) 2.11.ai_bg $24$ (not in LMFDB) 2.11.i_bg $24$ (not in LMFDB)