Properties

Label 1.7.c
Base field $\F_{7}$
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:  $\F_{7}$
Dimension:  $1$
L-polynomial:  $1 + 2 x + 7 x^{2}$
Frobenius angles:  $\pm0.623375857214$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-6}) \)
Galois group:  $C_2$
Jacobians:  2

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $10$ $60$ $310$ $2400$ $17050$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $10$ $60$ $310$ $2400$ $17050$ $117180$ $822790$ $5769600$ $40349290$ $282450300$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-6}) \).
All geometric endomorphisms are defined over $\F_{7}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.7.ac$2$1.49.k