Properties

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

Learn more about

Invariants

Base field:  $\F_{13}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 13 x^{2} )( 1 - 11 x + 55 x^{2} - 143 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.129998747777$, $\pm0.256122854178$, $\pm0.292104599859$
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
$A(\F_{q^r})$ 639 4601439 11225100852 23833986160959 51390434008460304 112449530624823290304 247013033277756500911683 542782277828473768235592975 1192549044869162504826166878804 2620020318392502099884099810974464

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 160 2323 29212 372773 4826557 62735440 815702932 10604639449 137859790175

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.af $\times$ 2.13.al_cd 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.ag_n_al$2$(not in LMFDB)
3.13.g_n_l$2$(not in LMFDB)
3.13.q_et_vp$2$(not in LMFDB)
3.13.an_dm_apg$3$(not in LMFDB)
3.13.ae_aj_dv$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ag_n_al$2$(not in LMFDB)
3.13.g_n_l$2$(not in LMFDB)
3.13.q_et_vp$2$(not in LMFDB)
3.13.an_dm_apg$3$(not in LMFDB)
3.13.ae_aj_dv$3$(not in LMFDB)
3.13.as_fp_azv$6$(not in LMFDB)
3.13.aj_bu_agu$6$(not in LMFDB)
3.13.e_aj_adv$6$(not in LMFDB)
3.13.j_bu_gu$6$(not in LMFDB)
3.13.n_dm_pg$6$(not in LMFDB)
3.13.s_fp_zv$6$(not in LMFDB)