Properties

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

Learn more about

Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.256122854178$
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
$A(\F_{q^r})$ 63 25137 4826304 819893529 137988113823 23293210300416 3936711602576151 665387401595286825 112455406959154934976 19005060581950480989777

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 148 2198 28708 371642 4825798 62737922 815694916 10604499374 137859194068

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 1.13.af 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 $\F_{13^{6}}$ is 1.4826809.atm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
All geometric endomorphisms are defined over $\F_{13^{6}}$.
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
2.13.ac_aj$2$2.169.aw_md
2.13.c_aj$2$2.169.aw_md
2.13.aj_bo$3$(not in LMFDB)
2.13.ad_q$3$(not in LMFDB)
2.13.a_ax$3$(not in LMFDB)
2.13.a_b$3$(not in LMFDB)
2.13.a_w$3$(not in LMFDB)
2.13.d_q$3$(not in LMFDB)
2.13.j_bo$3$(not in LMFDB)
2.13.m_cj$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ac_aj$2$2.169.aw_md
2.13.c_aj$2$2.169.aw_md
2.13.aj_bo$3$(not in LMFDB)
2.13.ad_q$3$(not in LMFDB)
2.13.a_ax$3$(not in LMFDB)
2.13.a_b$3$(not in LMFDB)
2.13.a_w$3$(not in LMFDB)
2.13.d_q$3$(not in LMFDB)
2.13.j_bo$3$(not in LMFDB)
2.13.m_cj$3$(not in LMFDB)
2.13.ao_cx$6$(not in LMFDB)
2.13.ak_bz$6$(not in LMFDB)
2.13.ah_bk$6$(not in LMFDB)
2.13.af_m$6$(not in LMFDB)
2.13.ae_be$6$(not in LMFDB)
2.13.a_w$6$(not in LMFDB)
2.13.e_be$6$(not in LMFDB)
2.13.f_m$6$(not in LMFDB)
2.13.h_bk$6$(not in LMFDB)
2.13.k_bz$6$(not in LMFDB)
2.13.o_cx$6$(not in LMFDB)
2.13.a_aw$12$(not in LMFDB)
2.13.a_ab$12$(not in LMFDB)
2.13.a_x$12$(not in LMFDB)