Properties

Label 2.17.aj_bs
Base Field $\F_{17}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{17}$
Dimension:  $2$
L-polynomial:  $1 - 9 x + 44 x^{2} - 153 x^{3} + 289 x^{4}$
Frobenius angles:  $\pm0.116336544503$, $\pm0.449669877836$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:  $C_2^2$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 172 85312 24143296 6931770624 2014449490252 582898741743616 168406571598364684 48661978739878425600 14063084451840828627904 4064236334561133871365952

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 9 297 4914 82993 1418769 24149022 410408721 6975870241 118587876498 2015996344857

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{41})\).
Endomorphism algebra over $\overline{\F}_{17}$
The base change of $A$ to $\F_{17^{6}}$ is 1.24137569.img 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$
All geometric endomorphisms are defined over $\F_{17^{6}}$.
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
2.17.j_bs$2$(not in LMFDB)
2.17.a_ah$3$(not in LMFDB)
2.17.j_bs$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.17.j_bs$2$(not in LMFDB)
2.17.a_ah$3$(not in LMFDB)
2.17.j_bs$3$(not in LMFDB)
2.17.a_h$12$(not in LMFDB)