Properties

Label 2.17.ai_bg
Base field $\F_{17}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
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:  $2$
Isomorphism classes:  3

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $178$ $82948$ $23402194$ $6880370704$ $2013144841618$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{4}}$.

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
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)