Properties

Label 2.13.ak_bv
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$
  $1 - 10 x + 47 x^{2} - 130 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.363422825076$
Angle rank:  $2$ (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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $77$ $27489$ $4868864$ $811722681$ $137321294957$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $164$ $2218$ $28420$ $369844$ $4822598$ $62752708$ $815806084$ $10604753554$ $137858748164$

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.ah $\times$ 1.13.ad 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.ae_f$2$2.169.ag_acb
2.13.e_f$2$2.169.ag_acb
2.13.k_bv$2$2.169.ag_acb
2.13.ab_u$3$(not in LMFDB)
2.13.c_l$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.13.ae_f$2$2.169.ag_acb
2.13.e_f$2$2.169.ag_acb
2.13.k_bv$2$2.169.ag_acb
2.13.ab_u$3$(not in LMFDB)
2.13.c_l$3$(not in LMFDB)
2.13.ai_bp$6$(not in LMFDB)
2.13.af_bg$6$(not in LMFDB)
2.13.ac_l$6$(not in LMFDB)
2.13.b_u$6$(not in LMFDB)
2.13.f_bg$6$(not in LMFDB)
2.13.i_bp$6$(not in LMFDB)