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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 328 1600640 2457351400 3182642147840 4168977020568448 5542992311720528000 7390812615085830357608 9846801822698305327319040 13109676769529277216077482600 17449575247994767399554560000000

Point counts of the (virtual) curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)