Properties

Label 2.17.ai_bg
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:  $1 - 8 x + 32 x^{2} - 136 x^{3} + 289 x^{4}$
Frobenius angles:  $\pm0.00936746300954$, $\pm0.509367463010$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{8})\)
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
$A(\F_{q^r})$ 178 82948 23402194 6880370704 2013144841618 582622191933700 168358154431328242 48658925715634520064 14063027005371573629746 4064231406644218986637828

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 290 4762 82374 1417850 24137570 410290730 6975432574 118587392074 2015993900450

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).
Endomorphism algebra over $\overline{\F}_{17}$
The base change of $A$ to $\F_{17^{4}}$ is 1.83521.awc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
All geometric endomorphisms are defined over $\F_{17^{4}}$.
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
2.17.i_bg$2$(not in LMFDB)
2.17.am_cs$8$(not in LMFDB)
2.17.a_ac$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.17.i_bg$2$(not in LMFDB)
2.17.am_cs$8$(not in LMFDB)
2.17.a_ac$8$(not in LMFDB)
2.17.a_c$8$(not in LMFDB)
2.17.m_cs$8$(not in LMFDB)
2.17.ag_t$24$(not in LMFDB)
2.17.g_t$24$(not in LMFDB)