Properties

Label 3.13.ap_dx_aqq
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 32 x^{2} - 104 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0370621216586$, $\pm0.0772104791556$, $\pm0.462937878341$
Angle rank:  $2$ (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
$A(\F_{q^r})$ 630 4154220 10012857120 22657863639600 50895254892840750 112443633371876002560 247075057715527390608630 542769149576876086635724800 1192514971681397866466643104160 2620006291359067191715284001015500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 147 2072 27767 369179 4826304 62751191 815683199 10604336456 137859052107

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bg and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{13}$
The base change of $A$ to $\F_{13^{4}}$ is 1.28561.alq 2 $\times$ 1.28561.ahj. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field

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.ab_al_q$2$(not in LMFDB)
3.13.b_al_aq$2$(not in LMFDB)
3.13.p_dx_qq$2$(not in LMFDB)
3.13.ag_bd_afo$3$(not in LMFDB)
3.13.ad_f_abw$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_al_q$2$(not in LMFDB)
3.13.b_al_aq$2$(not in LMFDB)
3.13.p_dx_qq$2$(not in LMFDB)
3.13.ag_bd_afo$3$(not in LMFDB)
3.13.ad_f_abw$3$(not in LMFDB)
3.13.an_dh_aoe$6$(not in LMFDB)
3.13.ak_cj_akm$6$(not in LMFDB)
3.13.d_f_bw$6$(not in LMFDB)
3.13.g_bd_fo$6$(not in LMFDB)
3.13.k_cj_km$6$(not in LMFDB)
3.13.n_dh_oe$6$(not in LMFDB)
3.13.ah_h_bq$8$(not in LMFDB)
3.13.ah_t_abq$8$(not in LMFDB)
3.13.h_h_abq$8$(not in LMFDB)
3.13.h_t_bq$8$(not in LMFDB)
3.13.af_h_be$24$(not in LMFDB)
3.13.af_t_abe$24$(not in LMFDB)
3.13.ac_h_m$24$(not in LMFDB)
3.13.ac_t_am$24$(not in LMFDB)
3.13.c_h_am$24$(not in LMFDB)
3.13.c_t_m$24$(not in LMFDB)
3.13.f_h_abe$24$(not in LMFDB)
3.13.f_t_be$24$(not in LMFDB)