Properties

Label 2.32.aq_ep
Base field $\F_{2^{5}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{5}}$
Dimension:  $2$
L-polynomial:  $( 1 - 11 x + 32 x^{2} )( 1 - 5 x + 32 x^{2} )$
  $1 - 16 x + 119 x^{2} - 512 x^{3} + 1024 x^{4}$
Frobenius angles:  $\pm0.0751336404065$, $\pm0.354289791955$
Angle rank:  $2$ (numerical)
Jacobians:  $25$
Isomorphism classes:  100

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $616$ $1029952$ $1076331256$ $1098806351104$ $1125459241405576$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $17$ $1007$ $32849$ $1047903$ $33541297$ $1073671247$ $34359745297$ $1099514101951$ $35184388166513$ $1125899945477807$

Jacobians and polarizations

This isogeny class contains the Jacobians of 25 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{5}}$.

Endomorphism algebra over $\F_{2^{5}}$
The isogeny class factors as 1.32.al $\times$ 1.32.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.32.ag_j$2$2.1024.as_agt
2.32.g_j$2$2.1024.as_agt
2.32.q_ep$2$2.1024.as_agt