# Properties

 Label 3.11.ap_ec_are 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: $( 1 - 4 x + 11 x^{2} )( 1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4} )$ Frobenius angles: $\pm0.0215640055172$, $\pm0.270299311731$, $\pm0.293962833700$ Angle rank: $3$ (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 $A(\F_{q^r})$ 328 1600640 2457351400 3182642147840 4168977020568448 5542992311720528000 7390812615085830357608 9846801822698305327319040 13109676769529277216077482600 17449575247994767399554560000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 109 1389 14849 160732 1766161 19462307 214295089 2357890599 25937681724

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ae $\times$ 2.11.al_bz and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ah_s_abm $2$ (not in LMFDB) 3.11.h_s_bm $2$ (not in LMFDB) 3.11.p_ec_re $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ah_s_abm $2$ (not in LMFDB) 3.11.h_s_bm $2$ (not in LMFDB) 3.11.p_ec_re $2$ (not in LMFDB) 3.11.af_g_o $5$ (not in LMFDB) 3.11.af_bk_aec $5$ (not in LMFDB) 3.11.a_b_cm $5$ (not in LMFDB) 3.11.f_q_bi $5$ (not in LMFDB) 3.11.an_dk_any $10$ (not in LMFDB) 3.11.ai_bh_aei $10$ (not in LMFDB) 3.11.af_q_abi $10$ (not in LMFDB) 3.11.ad_ac_cg $10$ (not in LMFDB) 3.11.ad_bc_ack $10$ (not in LMFDB) 3.11.a_b_acm $10$ (not in LMFDB) 3.11.d_ac_acg $10$ (not in LMFDB) 3.11.d_bc_ck $10$ (not in LMFDB) 3.11.f_g_ao $10$ (not in LMFDB) 3.11.f_bk_ec $10$ (not in LMFDB) 3.11.i_bh_ei $10$ (not in LMFDB) 3.11.n_dk_ny $10$ (not in LMFDB)