Properties

Label 1.25.g
Base field $\F_{5^{2}}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{5^{2}}$
Dimension:  $1$
L-polynomial:  $1 + 6 x + 25 x^{2}$
Frobenius angles:  $\pm0.704832764699$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1}) \)
Galois group:  $C_2$
Jacobians:  $4$
Isomorphism classes:  4

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $32$ $640$ $15392$ $391680$ $9765152$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $32$ $640$ $15392$ $391680$ $9765152$ $244117120$ $6103668512$ $152587560960$ $3814695421472$ $95367450947200$

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 $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{2}}$.

SubfieldPrimitive Model
$\F_{5}$1.5.ac
$\F_{5}$1.5.c

Twists

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

TwistExtension degreeCommon base change
1.25.ag$2$1.625.o
1.25.ai$4$(not in LMFDB)
1.25.i$4$(not in LMFDB)