# 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

## 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 329 1605191 2465610224 3196135929875 4184414508938279 5555032419146583296 7397480783624865461269 9849410116565608776796875 13110320776631729204269650704 17449630619462574846939784318391

 $r$ 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.
 Twist Extension Degree Common 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)