Properties

Label 2.13.ak_bz
Base Field $\F_{13}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 13 x^{2} )^{2}$
Frobenius angles:  $\pm0.256122854178$, $\pm0.256122854178$
Angle rank:  $1$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 81 29241 5143824 835152201 138435340761 23293210300416 3935794779943569 665324522896775625 112452896477996048016 19005002916235392383721

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 172 2338 29236 372844 4825798 62723308 815617828 10604262634 137858775772

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
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
2.13.a_b$2$2.169.c_nb
2.13.k_bz$2$2.169.c_nb
2.13.ah_bk$3$(not in LMFDB)
2.13.ae_be$3$(not in LMFDB)
2.13.c_aj$3$(not in LMFDB)
2.13.f_m$3$(not in LMFDB)
2.13.o_cx$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.a_b$2$2.169.c_nb
2.13.k_bz$2$2.169.c_nb
2.13.ah_bk$3$(not in LMFDB)
2.13.ae_be$3$(not in LMFDB)
2.13.c_aj$3$(not in LMFDB)
2.13.f_m$3$(not in LMFDB)
2.13.o_cx$3$(not in LMFDB)
2.13.a_ab$4$(not in LMFDB)
2.13.ao_cx$6$(not in LMFDB)
2.13.am_cj$6$(not in LMFDB)
2.13.aj_bo$6$(not in LMFDB)
2.13.af_m$6$(not in LMFDB)
2.13.ad_q$6$(not in LMFDB)
2.13.ac_aj$6$(not in LMFDB)
2.13.a_ax$6$(not in LMFDB)
2.13.a_w$6$(not in LMFDB)
2.13.d_q$6$(not in LMFDB)
2.13.e_be$6$(not in LMFDB)
2.13.h_bk$6$(not in LMFDB)
2.13.j_bo$6$(not in LMFDB)
2.13.m_cj$6$(not in LMFDB)
2.13.a_aw$12$(not in LMFDB)
2.13.a_x$12$(not in LMFDB)