Properties

Label 2.13.ak_by
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:  $( 1 - 6 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$
Frobenius angles:  $\pm0.187167041811$, $\pm0.312832958189$
Angle rank:  $1$ (numerical)
Jacobians:  5

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 5 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 80 28800 5074640 829440000 138211672400 23298078211200 3937112223300560 665417390653440000 112456038027485859920 19004963774689959120000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 170 2308 29038 372244 4826810 62744308 815731678 10604558884 137858491850

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ag $\times$ 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:
Endomorphism algebra over $\overline{\F}_{13}$
The base change of $A$ to $\F_{13^{4}}$ is 1.28561.je 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field

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_c$2$2.169.a_je
2.13.c_c$2$2.169.a_je
2.13.k_by$2$2.169.a_je
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.ac_c$2$2.169.a_je
2.13.c_c$2$2.169.a_je
2.13.k_by$2$2.169.a_je
2.13.am_ck$4$(not in LMFDB)
2.13.ai_bq$4$(not in LMFDB)
2.13.a_ak$4$(not in LMFDB)
2.13.a_k$4$(not in LMFDB)
2.13.i_bq$4$(not in LMFDB)
2.13.m_ck$4$(not in LMFDB)
2.13.a_ay$8$(not in LMFDB)
2.13.a_y$8$(not in LMFDB)
2.13.ag_x$12$(not in LMFDB)
2.13.ae_d$12$(not in LMFDB)
2.13.e_d$12$(not in LMFDB)
2.13.g_x$12$(not in LMFDB)