Properties

Label 3.11.ap_dz_aqk
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:  $( 1 - 6 x + 11 x^{2} )( 1 - 9 x + 38 x^{2} - 99 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0468428922585$, $\pm0.140218899004$, $\pm0.380176225592$
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
$A(\F_{q^r})$ 312 1505088 2325611808 3102311467008 4154340757959912 5556647549495577600 7403498659859443910808 9851799344685116613230592 13110361394916624828365551968 17449270344778098774861829700928

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 103 1314 14471 160167 1770520 19495725 214403855 2358013734 25937228503

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 2.11.aj_bm and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.acna 2 $\times$ 1.1771561.dly. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{11^{6}}$.
Remainder of endomorphism lattice by field

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.ad_af_be$2$(not in LMFDB)
3.11.d_af_abe$2$(not in LMFDB)
3.11.p_dz_qk$2$(not in LMFDB)
3.11.ag_q_abe$3$(not in LMFDB)
3.11.d_af_abe$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ad_af_be$2$(not in LMFDB)
3.11.d_af_abe$2$(not in LMFDB)
3.11.p_dz_qk$2$(not in LMFDB)
3.11.ag_q_abe$3$(not in LMFDB)
3.11.d_af_abe$3$(not in LMFDB)
3.11.ag_q_abe$6$(not in LMFDB)
3.11.ad_af_be$6$(not in LMFDB)
3.11.g_q_be$6$(not in LMFDB)
3.11.ag_g_be$12$(not in LMFDB)
3.11.g_g_abe$12$(not in LMFDB)