Properties

Label 2.13.am_cj
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )$
  $1 - 12 x + 61 x^{2} - 156 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.256122854178$
Angle rank:  $1$ (numerical)
Jacobians:  $2$
Isomorphism classes:  4

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $63$ $25137$ $4826304$ $819893529$ $137988113823$

Point counts of the curve

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

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 $\F_{13^{6}}$.

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
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)