Properties

Label 1.4.ae
Base field $\F_{2^{2}}$
Dimension $1$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $1$
L-polynomial:  $( 1 - 2 x )^{2}$
  $1 - 4x + 4x^{2}$
Frobenius angles:  $0$, $0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q\)
Galois group:  Trivial
Jacobians:  1

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $9$ $49$ $225$ $961$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $9$ $49$ $225$ $961$ $3969$ $16129$ $65025$ $261121$ $1046529$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
1.4.e$2$1.16.ai
1.4.c$3$1.64.aq
1.4.a$4$1.256.abg
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.4.e$2$1.16.ai
1.4.c$3$1.64.aq
1.4.a$4$1.256.abg
1.4.ac$6$(not in LMFDB)

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.