Properties

Label 2.113.abk_uz
Base Field $\F_{113}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{113}$
Dimension:  $2$
L-polynomial:  $1 - 36 x + 545 x^{2} - 4068 x^{3} + 12769 x^{4}$
Frobenius angles:  $\pm0.0992057967989$, $\pm0.234127536534$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:  $C_2^2$
Jacobians:  22

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 22 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 9211 160446409 2081952603184 26587197225882441 339460976702890640491 4334526641904634166937856 55347526555654103051969171371 706732551601732348919733293101449 9024267965168278303972721530516379056 115230877653756324445990574469069096754249

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 78 12564 1442898 163064260 18424581798 2081953453758 235260552675030 26584441888512004 3004041937984268274 339456739011437071764

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{113}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).
Endomorphism algebra over $\overline{\F}_{113}$
The base change of $A$ to $\F_{113^{6}}$ is 1.2081951752609.bwkgk 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
All geometric endomorphisms are defined over $\F_{113^{6}}$.
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
2.113.a_hy$3$(not in LMFDB)
2.113.bk_uz$3$(not in LMFDB)
2.113.a_hy$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.113.a_hy$3$(not in LMFDB)
2.113.bk_uz$3$(not in LMFDB)
2.113.a_hy$6$(not in LMFDB)
2.113.a_ahy$12$(not in LMFDB)