Properties

Label 3.13.ap_ee_asm
Base Field $\F_{13}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $( 1 - 6 x + 13 x^{2} )( 1 - 9 x + 41 x^{2} - 117 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.109149799241$, $\pm0.187167041811$, $\pm0.400911184348$
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
$A(\F_{q^r})$ 680 4583200 10829381480 23350945680000 51206443416310400 112568218669671623200 247196800994063115498920 542866357155795385896000000 1192536010021171371049232913320 2619982089030695746857167994880000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 161 2243 28625 371444 4831649 62782103 815829281 10604523539 137857778636

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ag $\times$ 2.13.aj_bp 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 $\F_{13}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ad_a_m$2$(not in LMFDB)
3.13.d_a_am$2$(not in LMFDB)
3.13.p_ee_sm$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ad_a_m$2$(not in LMFDB)
3.13.d_a_am$2$(not in LMFDB)
3.13.p_ee_sm$2$(not in LMFDB)
3.13.an_dm_api$4$(not in LMFDB)
3.13.af_s_acs$4$(not in LMFDB)
3.13.f_s_cs$4$(not in LMFDB)
3.13.n_dm_pi$4$(not in LMFDB)