Properties

Label 2.13.al_cc
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

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$
  $1 - 11 x + 54 x^{2} - 143 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.312832958189$
Angle rank:  $2$ (numerical)
Jacobians:  $1$
Isomorphism classes:  9

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $70$ $26460$ $4873120$ $817084800$ $137610200350$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $157$ $2220$ $28609$ $370623$ $4822234$ $62737419$ $815751841$ $10604790060$ $137859735157$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is 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.ah $\times$ 1.13.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

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.ad_ac$2$2.169.an_ee
2.13.d_ac$2$2.169.an_ee
2.13.l_cc$2$2.169.an_ee
2.13.ac_s$3$(not in LMFDB)
2.13.b_g$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.13.ad_ac$2$2.169.an_ee
2.13.d_ac$2$2.169.an_ee
2.13.l_cc$2$2.169.an_ee
2.13.ac_s$3$(not in LMFDB)
2.13.b_g$3$(not in LMFDB)
2.13.an_cq$4$(not in LMFDB)
2.13.ab_aq$4$(not in LMFDB)
2.13.b_aq$4$(not in LMFDB)
2.13.n_cq$4$(not in LMFDB)
2.13.aj_bu$6$(not in LMFDB)
2.13.ag_bi$6$(not in LMFDB)
2.13.ab_g$6$(not in LMFDB)
2.13.c_s$6$(not in LMFDB)
2.13.g_bi$6$(not in LMFDB)
2.13.j_bu$6$(not in LMFDB)
2.13.al_ce$12$(not in LMFDB)
2.13.ai_bm$12$(not in LMFDB)
2.13.ae_o$12$(not in LMFDB)
2.13.ab_ae$12$(not in LMFDB)
2.13.b_ae$12$(not in LMFDB)
2.13.e_o$12$(not in LMFDB)
2.13.i_bm$12$(not in LMFDB)
2.13.l_ce$12$(not in LMFDB)