Properties

Label 1.169.aba
Base field $\F_{13^{2}}$
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:  $\F_{13^{2}}$
Dimension:  $1$
L-polynomial:  $( 1 - 13 x )^{2}$
  $1 - 26 x + 169 x^{2}$
Frobenius angles:  $0$, $0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q\)
Galois group:  Trivial
Jacobians:  $1$

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

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $144$ $28224$ $4822416$ $815673600$ $137857749264$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $144$ $28224$ $4822416$ $815673600$ $137857749264$ $23298075468864$ $3937376260202256$ $665416607551718400$ $112455406930748394384$ $19004963774605082455104$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
1.169.ba$2$(not in LMFDB)