# Properties

 Label 2.11.af_o 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: $1 - 5 x + 14 x^{2} - 55 x^{3} + 121 x^{4}$ Frobenius angles: $\pm0.105104110453$, $\pm0.561562556214$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Galois group: $C_2^2$ Jacobians: 7

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 7 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 76 14896 1669264 210986944 26054551876 3142195845376 379697887884916 45955090747125504 5560368732876601744 672755273810511468976

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 125 1252 14409 161777 1773686 19484507 214383889 2358139132 25937628125

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{-19})$$.
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{3}}$ is 1.1331.abo 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-19})$$$)$
All geometric endomorphisms are defined over $\F_{11^{3}}$.

## 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.f_o $2$ 2.121.d_aei 2.11.k_bv $3$ (not in LMFDB) 2.11.ak_bv $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.f_o $2$ 2.121.d_aei 2.11.k_bv $3$ (not in LMFDB) 2.11.ak_bv $6$ (not in LMFDB) 2.11.a_ad $6$ (not in LMFDB) 2.11.f_o $6$ (not in LMFDB) 2.11.a_d $12$ (not in LMFDB)