Properties

Label 3.11.ap_ec_ard
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:  $( 1 - 5 x + 11 x^{2} )( 1 - 10 x + 45 x^{2} - 110 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0820279942768$, $\pm0.228229222880$, $\pm0.318205720493$
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
$A(\F_{q^r})$ 329 1605191 2465610224 3196135929875 4184414508938279 5555032419146583296 7397480783624865461269 9849410116565608776796875 13110320776631729204269650704 17449630619462574846939784318391

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 109 1392 14909 161327 1770004 19479877 214351861 2358006432 25937764029

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.af $\times$ 2.11.ak_bt 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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.af_g_f$2$(not in LMFDB)
3.11.f_g_af$2$(not in LMFDB)
3.11.p_ec_rd$2$(not in LMFDB)