Properties

Label 3.13.ap_dz_are
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 34 x^{2} - 104 x^{3} + 169 x^{4} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.104164352389$, $\pm0.448054596667$
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
$A(\F_{q^r})$ 644 4273584 10238497472 22856905042944 51016152313775124 112539965532274732032 247170331084403268714116 542837458761077365592211456 1192546278982659896218419027392 2620019421645100182730662587955504

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 151 2120 28015 370059 4830436 62775383 815785855 10604614856 137859742991

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bi 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.ab_aj_be$2$(not in LMFDB)
3.13.b_aj_abe$2$(not in LMFDB)
3.13.p_dz_re$2$(not in LMFDB)
3.13.ag_bf_afk$3$(not in LMFDB)
3.13.ad_h_abm$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.ab_aj_be$2$(not in LMFDB)
3.13.b_aj_abe$2$(not in LMFDB)
3.13.p_dz_re$2$(not in LMFDB)
3.13.ag_bf_afk$3$(not in LMFDB)
3.13.ad_h_abm$3$(not in LMFDB)
3.13.an_dj_aoo$6$(not in LMFDB)
3.13.ak_cl_akq$6$(not in LMFDB)
3.13.d_h_bm$6$(not in LMFDB)
3.13.g_bf_fk$6$(not in LMFDB)
3.13.k_cl_kq$6$(not in LMFDB)
3.13.n_dj_oo$6$(not in LMFDB)