Properties

Label 2.13.aj_bu
Base Field $\F_{13}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$
Frobenius angles:  $\pm0.256122854178$, $\pm0.312832958189$
Angle rank:  $2$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 90 30780 5193720 832291200 138056202450 23276009352960 3935763223779330 665370955154880000 112455979023092576760 19005077510025702543900

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 181 2360 29137 371825 4822234 62722805 815674753 10604553320 137859316861

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.af $\times$ 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:
All geometric endomorphisms are defined over $\F_{13}$.

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_g$2$2.169.l_nk
2.13.b_g$2$2.169.l_nk
2.13.j_bu$2$2.169.l_nk
2.13.ag_bi$3$(not in LMFDB)
2.13.d_ac$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ab_g$2$2.169.l_nk
2.13.b_g$2$2.169.l_nk
2.13.j_bu$2$2.169.l_nk
2.13.ag_bi$3$(not in LMFDB)
2.13.d_ac$3$(not in LMFDB)
2.13.al_ce$4$(not in LMFDB)
2.13.ab_ae$4$(not in LMFDB)
2.13.b_ae$4$(not in LMFDB)
2.13.l_ce$4$(not in LMFDB)
2.13.al_cc$6$(not in LMFDB)
2.13.ad_ac$6$(not in LMFDB)
2.13.ac_s$6$(not in LMFDB)
2.13.c_s$6$(not in LMFDB)
2.13.g_bi$6$(not in LMFDB)
2.13.l_cc$6$(not in LMFDB)
2.13.an_cq$12$(not in LMFDB)
2.13.ai_bm$12$(not in LMFDB)
2.13.ae_o$12$(not in LMFDB)
2.13.ab_aq$12$(not in LMFDB)
2.13.b_aq$12$(not in LMFDB)
2.13.e_o$12$(not in LMFDB)
2.13.i_bm$12$(not in LMFDB)
2.13.n_cq$12$(not in LMFDB)