Abelian variety isogeny class 1.73.ac over $\F_{73}$

• Label: 1.73.ac
• Base field: $\F_{73}$
• 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:	$\F_{73}$
Dimension:	$1$
L-polynomial:	$1 - 2 x + 73 x^{2}$
Frobenius angles:	$\pm0.462659059226$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}) \)
Galois group:	$C_2$
Jacobians:	$9$
Isomorphism classes:	9

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$72$	$5472$	$389448$	$28388736$	$2073021192$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$5472$	$389448$	$28388736$	$2073021192$	$151334819424$	$11047403384712$	$806460058326528$	$58871586285942984$	$4297625831309339232$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which 0 are hyperelliptic):

• $y^2=x^3+23 x+23$
• $y^2=x^3+55 x+56$
• $y^2=x^3+49 x+26$
• $y^2=x^3+27 x+27$
• $y^2=x^3+65 x+33$
• $y^2=x^3+41 x+41$
• $y^2=x^3+20 x+20$
• $y^2=x^3+14 x+70$
• $y^2=x^3+x+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73}$.

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \).

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
1.73.c	$2$	(not in LMFDB)