Properties

Label 1.625.aby
Base field $\F_{5^{4}}$
Dimension $1$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple yes
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $1$
L-polynomial:  $( 1 - 25 x )^{2}$
  $1 - 50 x + 625 x^{2}$
Frobenius angles:  $0$, $0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q\)
Galois group:  Trivial

This isogeny class is simple and geometrically simple, not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $576$ $389376$ $244109376$ $152587109376$ $95367412109376$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $576$ $389376$ $244109376$ $152587109376$ $95367412109376$ $59604644287109376$ $37252902972412109376$ $23283064365081787109376$ $14551915228359222412109376$ $9094947017729091644287109376$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{4}}$.

SubfieldPrimitive Model
$\F_{5}$1.5.a

Twists

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

TwistExtension degreeCommon base change
1.625.by$2$(not in LMFDB)
1.625.z$3$(not in LMFDB)
1.625.az$6$(not in LMFDB)