Properties

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

Learn more about

Invariants

Base field:  $\F_{11}$
Dimension:  $3$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )^{2}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.228229222880$, $\pm0.228229222880$
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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 294 1529388 2473452576 3249949500000 4227425799974454 5573941210112486400 7401573355048140841314 9848682797040213462000000 13109296997460963836814159456 17449193974096987274709084680748

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 102 1394 15154 162976 1776024 19490656 214336034 2357822294 25937114982

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 1.11.af 2 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.ag_i_s$2$(not in LMFDB)
3.11.ae_ac_ck$2$(not in LMFDB)
3.11.e_ac_ack$2$(not in LMFDB)
3.11.g_i_as$2$(not in LMFDB)
3.11.q_eo_ti$2$(not in LMFDB)
3.11.ab_af_ba$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ag_i_s$2$(not in LMFDB)
3.11.ae_ac_ck$2$(not in LMFDB)
3.11.e_ac_ack$2$(not in LMFDB)
3.11.g_i_as$2$(not in LMFDB)
3.11.q_eo_ti$2$(not in LMFDB)
3.11.ab_af_ba$3$(not in LMFDB)
3.11.ag_o_as$4$(not in LMFDB)
3.11.g_o_s$4$(not in LMFDB)
3.11.al_cd_ahm$6$(not in LMFDB)
3.11.b_af_aba$6$(not in LMFDB)
3.11.l_cd_hm$6$(not in LMFDB)