Abelian variety isogeny class 1.125.ae over $\F_{5^{3}}$

• Label: 1.125.ae
• Base field: $\F_{5^{3}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{3}}$
Dimension:	$1$
L-polynomial:	$1 - 4 x + 125 x^{2}$
Frobenius angles:	$\pm0.442750852951$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-1}) \)
Galois group:	$C_2$
Jacobians:	$7$
Isomorphism classes:	7

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$122$	$15860$	$1954562$	$244117120$	$30517304602$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$15860$	$1954562$	$244117120$	$30517304602$	$3814699109780$	$476837199770242$	$59604644711139840$	$7450580591470935482$	$931322574601698293300$

Jacobians and polarizations

This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{3}}$.

Endomorphism algebra over $\F_{5^{3}}$

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 $\F_{5^{3}}$.

Subfield	Primitive Model
$\F_{5}$	1.5.ae

Twists

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

Twist	Extension degree	Common base change
1.125.e	$2$	(not in LMFDB)
1.125.aw	$4$	(not in LMFDB)
1.125.w	$4$	(not in LMFDB)