Properties

Label 2.17.am_cn
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:  $1 - 12 x + 65 x^{2} - 204 x^{3} + 289 x^{4}$
Frobenius angles:  $\pm0.0157896134134$, $\pm0.349122946747$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:  $C_2^2$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 139 79369 24128176 6943914441 2010360903259 582168877086976 168359278701453979 48661044555091420809 14063084451918975425584 4064226529530741241632649

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 276 4914 83140 1415886 24118782 410293470 6975736324 118587876498 2015991481236

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).
Endomorphism algebra over $\overline{\F}_{17}$
The base change of $A$ to $\F_{17^{6}}$ is 1.24137569.anxi 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
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.m_cn$2$(not in LMFDB)
2.17.a_o$3$(not in LMFDB)
2.17.m_cn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.17.m_cn$2$(not in LMFDB)
2.17.a_o$3$(not in LMFDB)
2.17.m_cn$3$(not in LMFDB)
2.17.a_o$6$(not in LMFDB)
2.17.a_ao$12$(not in LMFDB)