Properties

Label 1.67.aq
Base field $\F_{67}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{67}$
Dimension:  $1$
L-polynomial:  $1 - 16 x + 67 x^{2}$
Frobenius angles:  $\pm0.0678686046652$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  $2$
Isomorphism classes:  2

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $52$ $4368$ $299884$ $20145216$ $1350089572$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $52$ $4368$ $299884$ $20145216$ $1350089572$ $90458209296$ $6060711220252$ $406067682978048$ $27206534508837268$ $1822837805989204368$

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 $\F_{67}$.

Endomorphism algebra over $\F_{67}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
1.67.q$2$(not in LMFDB)
1.67.f$3$(not in LMFDB)
1.67.l$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
1.67.q$2$(not in LMFDB)
1.67.f$3$(not in LMFDB)
1.67.l$3$(not in LMFDB)
1.67.al$6$(not in LMFDB)
1.67.af$6$(not in LMFDB)