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

• Label: 1.625.g
• 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:	$1 + 6 x + 625 x^{2}$
Frobenius angles:	$\pm0.538289458774$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-154}) \)
Galois group:	$C_2$
Jacobians:	$24$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$632$	$391840$	$244129592$	$152587198080$	$95367442692152$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$632$	$391840$	$244129592$	$152587198080$	$95367442692152$	$59604645141922720$	$37252902975512744312$	$23283064365212518709760$	$14551915228373589969422072$	$9094947017729350977786743200$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which 0 are hyperelliptic):

• $y^2=x^3+a^{358} x+a^{358}$
• $y^2=x^3+a^{478} x+a^{479}$
• $y^2=x^3+a^{82} x+a^{83}$
• $y^2=x^3+a^{622} x+a^{623}$
• $y^2=x^3+a^{214} x+a^{214}$
• $y^2=x^3+a^{130} x+a^{131}$
• $y^2=x^3+a^{14} x+a^{15}$
• $y^2=x^3+a^{46} x+a^{47}$
• $y^2=x^3+a^{126} x+a^{127}$
• $y^2=x^3+a^{542} x+a^{542}$
• $y^2=x^3+a^{302} x+a^{303}$
• $y^2=x^3+a^{254} x+a^{255}$
• $y^2=x^3+a^{510} x+a^{511}$
• $y^2=x^3+a^{414} x+a^{414}$
• $y^2=x^3+a^{62} x+a^{63}$
• $y^2=x^3+a^{582} x+a^{582}$
• $y^2=x^3+a^{198} x+a^{198}$
• $y^2=x^3+a^{446} x+a^{446}$
• $y^2=x^3+a^{222} x+a^{223}$
• $y^2=x^3+a^{526} x+a^{527}$
• $y^2=x^3+a^{606} x+a^{607}$
• $y^2=x^3+a^{366} x+a^{366}$
• $y^2=x^3+a^{402} x+a^{403}$
• $y^2=x^3+a^{18} x+a^{19}$

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 \(\Q(\sqrt{-154}) \).

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.ag	$2$	(not in LMFDB)