Abelian variety isogeny class 1.127.au over $\F_{127}$

• Label: 1.127.au
• Base field: $\F_{127}$
• 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:	$\F_{127}$
Dimension:	$1$
L-polynomial:	$1 - 20 x + 127 x^{2}$
Frobenius angles:	$\pm0.152539311644$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$6$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$108$	$15984$	$2048004$	$260155584$	$33038636508$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$108$	$15984$	$2048004$	$260155584$	$33038636508$	$4195876867056$	$532875905291124$	$67675234641580800$	$8594754750889682508$	$1091533853068127860464$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which 0 are hyperelliptic):

• $y^2=x^3+2 x+2$
• $y^2=x^3+99 x+99$
• $y^2=x^3+77 x+77$
• $y^2=x^3+54 x+54$
• $y^2=x^3+75 x+98$
• $y^2=x^3+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{127}$.

Endomorphism algebra over $\F_{127}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
1.127.u	$2$	(not in LMFDB)
1.127.b	$3$	(not in LMFDB)
1.127.t	$3$	(not in LMFDB)