Properties

Label 2.13.ai_bp
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:  $( 1 - 5 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$
Frobenius angles:  $\pm0.256122854178$, $\pm0.363422825076$
Angle rank:  $2$ (numerical)
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 99 31977 5189184 826829289 137766360699 23277766127616 3936722393153187 665415199774578825 112455591899559644736 19004941444519141523577

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 188 2358 28948 371046 4822598 62738094 815728996 10604516814 137858329868

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.af $\times$ 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:
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.ac_l$2$2.169.s_nr
2.13.c_l$2$2.169.s_nr
2.13.i_bp$2$2.169.s_nr
2.13.af_bg$3$(not in LMFDB)
2.13.e_f$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ac_l$2$2.169.s_nr
2.13.c_l$2$2.169.s_nr
2.13.i_bp$2$2.169.s_nr
2.13.af_bg$3$(not in LMFDB)
2.13.e_f$3$(not in LMFDB)
2.13.ak_bv$6$(not in LMFDB)
2.13.ae_f$6$(not in LMFDB)
2.13.ab_u$6$(not in LMFDB)
2.13.b_u$6$(not in LMFDB)
2.13.f_bg$6$(not in LMFDB)
2.13.k_bv$6$(not in LMFDB)