Properties

Label 2.13.ak_bz
Base field $\F_{13}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 13 x^{2} )^{2}$
  $1 - 10 x + 51 x^{2} - 130 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.256122854178$, $\pm0.256122854178$
Angle rank:  $1$ (numerical)
Jacobians:  $2$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $81$ $29241$ $5143824$ $835152201$ $138435340761$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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)