Properties

Label 1.169.ax
Base Field $\F_{13^{2}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $1$
L-polynomial:  $1 - 23 x + 169 x^{2}$
Frobenius angles:  $\pm0.154420958311$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 147 28371 4826304 815751363 137859052107 23298094520064 3937376507160243 665416610388590403 112455406959154934976 19004963774842628516451

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 147 28371 4826304 815751363 137859052107 23298094520064 3937376507160243 665416610388590403 112455406959154934976 19004963774842628516451

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{13^{2}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.

SubfieldPrimitive Model
$\F_{13}$1.13.ah
$\F_{13}$1.13.h

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.169.x$2$(not in LMFDB)
1.169.b$3$(not in LMFDB)
1.169.w$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.169.x$2$(not in LMFDB)
1.169.b$3$(not in LMFDB)
1.169.w$3$(not in LMFDB)
1.169.aw$6$(not in LMFDB)
1.169.ab$6$(not in LMFDB)