# Properties

 Label 3.11.aq_en_atc Base Field $\F_{11}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{11}$ Dimension: $3$ L-polynomial: $( 1 - 4 x + 11 x^{2} )( 1 - 6 x + 11 x^{2} )^{2}$ Frobenius angles: $\pm0.140218899004$, $\pm0.140218899004$, $\pm0.293962833700$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 288 1492992 2417234400 3203268083712 4203934039866528 5568437853216000000 7403646096458768439648 9851624936637438269325312 13110967172879088971994093600 17449746167746488620161164183552

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 100 1364 14940 162076 1774276 19496116 214400060 2358122684 25937935780

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag 2 $\times$ 1.11.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.11.ag 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.11.ae : $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{11}$.

## 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 3.11.ai_v_abg $2$ (not in LMFDB) 3.11.ae_ad_ce $2$ (not in LMFDB) 3.11.e_ad_ace $2$ (not in LMFDB) 3.11.i_v_bg $2$ (not in LMFDB) 3.11.q_en_tc $2$ (not in LMFDB) 3.11.c_m_bg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ai_v_abg $2$ (not in LMFDB) 3.11.ae_ad_ce $2$ (not in LMFDB) 3.11.e_ad_ace $2$ (not in LMFDB) 3.11.i_v_bg $2$ (not in LMFDB) 3.11.q_en_tc $2$ (not in LMFDB) 3.11.c_m_bg $3$ (not in LMFDB) 3.11.ae_z_ace $4$ (not in LMFDB) 3.11.e_z_ce $4$ (not in LMFDB) 3.11.ak_ci_aiy $6$ (not in LMFDB) 3.11.ac_m_abg $6$ (not in LMFDB) 3.11.k_ci_iy $6$ (not in LMFDB) 3.11.ai_bj_aeq $8$ (not in LMFDB) 3.11.a_d_ace $8$ (not in LMFDB) 3.11.a_d_ce $8$ (not in LMFDB) 3.11.i_bj_eq $8$ (not in LMFDB)