# Properties

 Label 2.11.ag_z 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: $1 - 6 x + 25 x^{2} - 66 x^{3} + 121 x^{4}$ Frobenius angles: $\pm0.193114434330$, $\pm0.473552232337$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Galois group: $C_2^2$ Jacobians: 11

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 11 curves, and hence is principally polarizable:

• $y^2=10x^6+8x^5+4x^4+x^3+6x^2+6x+5$
• $y^2=3x^6+6x^5+8x^4+5x^3+5x^2+10x+4$
• $y^2=2x^6+2x^5+3x^3+7x^2+8x+2$
• $y^2=10x^6+3x^5+8x^4+5x^3+7x^2+3x+2$
• $y^2=6x^6+9x^5+8x^4+x^3+5x^2+10x+3$
• $y^2=8x^6+4x^4+2x^3+8x^2+2x+5$
• $y^2=6x^6+2x^5+x^4+2x^3+10x^2+8x+10$
• $y^2=10x^6+6x^5+4x^4+7x^2+2$
• $y^2=8x^6+2x^5+3x^4+3x^3+9x^2+x+10$
• $y^2=2x^6+8x^5+5x^4+2x^3+5x^2+7x+10$
• $y^2=2x^6+4x^5+4x^2+7x+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 75 16425 1822500 213672825 26013826875 3146721210000 379921651716675 45943907113816425 5559605875406722500 672747467936585135625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 136 1368 14596 161526 1776238 19495986 214331716 2357815608 25937327176

## 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 $\F_{11^{3}}$ is 1.1331.s 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
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.g_z $2$ 2.121.o_cx 2.11.m_cg $3$ (not in LMFDB) 2.11.am_cg $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.11.g_z $2$ 2.121.o_cx 2.11.m_cg $3$ (not in LMFDB) 2.11.am_cg $6$ (not in LMFDB) 2.11.a_ao $6$ (not in LMFDB) 2.11.a_o $12$ (not in LMFDB) 2.11.ae_i $24$ (not in LMFDB) 2.11.e_i $24$ (not in LMFDB)