Properties

Label 2.13.ai_bg
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:  $1 - 8 x + 32 x^{2} - 104 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.0370621216586$, $\pm0.462937878341$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{10})\)
Galois group:  $C_2^2$
Jacobians:  3

This isogeny class is simple but not geometrically 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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 90 28260 4705290 798627600 137233172250 23298083455140 3937418050935690 665361008507289600 112452424073098948890 19004963775070323766500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 170 2142 27958 369606 4826810 62749182 815662558 10604218086 137858491850

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{10})\).
Endomorphism algebra over $\overline{\F}_{13}$
The base change of $A$ to $\F_{13^{4}}$ is 1.28561.alq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$
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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.13.i_bg$2$2.169.a_alq
2.13.a_ag$8$(not in LMFDB)
2.13.a_g$8$(not in LMFDB)