Properties

Label 2.5.ab_a
Base field $\F_{5}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - x - 5 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.189652078905$, $\pm0.706462326177$
Angle rank:  $2$ (numerical)
Number field:  4.0.8405.1
Galois group:  $D_{4}$
Jacobians:  $4$
Isomorphism classes:  4

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $20$ $640$ $13760$ $442880$ $10080100$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $25$ $110$ $705$ $3225$ $15670$ $78965$ $389985$ $1950230$ $9765825$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is 4.0.8405.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.5.b_a$2$2.25.ab_bo