Properties

Label 2.13.an_cq
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 no

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 - 6 x + 13 x^{2} )$
  $1 - 13 x + 68 x^{2} - 169 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.187167041811$
Angle rank:  $2$ (numerical)
Jacobians:  $0$
Isomorphism classes:  2

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $56$ $23520$ $4715648$ $817084800$ $138143507096$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $137$ $2146$ $28609$ $372061$ $4830374$ $62759425$ $815751841$ $10604504938$ $137858369057$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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.ag 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.ab_aq$2$2.169.abh_vw
2.13.b_aq$2$2.169.abh_vw
2.13.n_cq$2$2.169.abh_vw
2.13.ae_o$3$(not in LMFDB)
2.13.ab_ae$3$(not in LMFDB)

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

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