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

• Label: 1.73.ae
• 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:	$1 - 4 x + 73 x^{2}$
Frobenius angles:	$\pm0.424791481369$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-69}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$70$	$5460$	$389830$	$28392000$	$2072987350$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$5460$	$389830$	$28392000$	$2072987350$	$151334344980$	$11047405143670$	$806460109728000$	$58871586296009830$	$4297625826752649300$

Jacobians and polarizations

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

• $y^2=x^3+43 x+69$
• $y^2=x^3+35 x+35$
• $y^2=x^3+9 x+9$
• $y^2=x^3+43 x+43$
• $y^2=x^3+27 x+62$
• $y^2=x^3+60 x+8$
• $y^2=x^3+63 x+23$
• $y^2=x^3+66 x+38$

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 \(\Q(\sqrt{-69}) \).

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.e	$2$	(not in LMFDB)