Abelian variety isogeny class 1.625.aj over $\F_{5^{4}}$

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

Invariants

Base field:	$\F_{5^{4}}$
Dimension:	$1$
L-polynomial:	$1 - 9 x + 625 x^{2}$
Frobenius angles:	$\pm0.442390223303$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2419}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

This isogeny class is simple and geometrically simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$617$	$391795$	$244156772$	$152587305315$	$95367416281577$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$617$	$391795$	$244156772$	$152587305315$	$95367416281577$	$59604645002978560$	$37252902996266837657$	$23283064365349549705155$	$14551915228359235277327012$	$9094947017729237213627487475$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):

• $y^2=x^3+a^{162} x+a^{163}$
• $y^2=x^3+a^{34} x+a^{35}$
• $y^2=x^3+a^{178} x+a^{179}$
• $y^2=x^3+a^{610} x+a^{611}$
• $y^2=x^3+a^{498} x+a^{499}$
• $y^2=x^3+a^{258} x+a^{259}$
• $y^2=x^3+a^{546} x+a^{547}$
• $y^2=x^3+a^{226} x+a^{227}$

where $a$ is a root of the Conway polynomial.

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 \(\Q(\sqrt{-2419}) \).

Base change

This is a primitive isogeny class.

Twists

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

Twist	Extension degree	Common base change
1.625.j	$2$	(not in LMFDB)