Properties

Label 2.13.ai_bm
Base Field $\F_{13}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 13 x^{2} )( 1 - 2 x + 13 x^{2} )$
Frobenius angles:  $\pm0.187167041811$, $\pm0.410543812489$
Angle rank:  $2$ (numerical)
Jacobians:  12

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 contains the Jacobians of 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 96 30720 5025888 818380800 137854828896 23315295467520 3938851984386912 665446208805273600 112452955480771954272 19004772804678878361600

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 182 2286 28654 371286 4830374 62772030 815767006 10604268198 137857106582

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ag $\times$ 1.13.ac 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.ae_o$2$2.169.m_eo
2.13.e_o$2$2.169.m_eo
2.13.i_bm$2$2.169.m_eo
2.13.al_ce$3$(not in LMFDB)
2.13.b_aq$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ae_o$2$2.169.m_eo
2.13.e_o$2$2.169.m_eo
2.13.i_bm$2$2.169.m_eo
2.13.al_ce$3$(not in LMFDB)
2.13.b_aq$3$(not in LMFDB)
2.13.ag_bi$4$(not in LMFDB)
2.13.ac_s$4$(not in LMFDB)
2.13.c_s$4$(not in LMFDB)
2.13.g_bi$4$(not in LMFDB)
2.13.an_cq$6$(not in LMFDB)
2.13.ab_aq$6$(not in LMFDB)
2.13.ab_ae$6$(not in LMFDB)
2.13.b_ae$6$(not in LMFDB)
2.13.l_ce$6$(not in LMFDB)
2.13.n_cq$6$(not in LMFDB)
2.13.al_cc$12$(not in LMFDB)
2.13.aj_bu$12$(not in LMFDB)
2.13.ad_ac$12$(not in LMFDB)
2.13.ab_g$12$(not in LMFDB)
2.13.b_g$12$(not in LMFDB)
2.13.d_ac$12$(not in LMFDB)
2.13.j_bu$12$(not in LMFDB)
2.13.l_cc$12$(not in LMFDB)