Properties

Label 3.11.ao_ds_api
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:  $( 1 - 6 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )( 1 - 3 x + 11 x^{2} )$
Frobenius angles:  $\pm0.140218899004$, $\pm0.228229222880$, $\pm0.350615407277$
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
$A(\F_{q^r})$ 378 1735020 2531142432 3214992060000 4193507513145258 5562671473666944000 7402105932333456681918 9850932518460791934960000 13110305954076126290290434912 17449326704437058315563297195500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 118 1426 14994 161678 1772440 19492058 214384994 2358003766 25937312278

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 1.11.af $\times$ 1.11.ad 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.ai_be_adi$2$(not in LMFDB)
3.11.ae_g_c$2$(not in LMFDB)
3.11.ac_a_bu$2$(not in LMFDB)
3.11.c_a_abu$2$(not in LMFDB)
3.11.e_g_ac$2$(not in LMFDB)
3.11.i_be_di$2$(not in LMFDB)
3.11.o_ds_pi$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ai_be_adi$2$(not in LMFDB)
3.11.ae_g_c$2$(not in LMFDB)
3.11.ac_a_bu$2$(not in LMFDB)
3.11.c_a_abu$2$(not in LMFDB)
3.11.e_g_ac$2$(not in LMFDB)
3.11.i_be_di$2$(not in LMFDB)
3.11.o_ds_pi$2$(not in LMFDB)