Abelian variety isogeny class 1.49.ab over $\F_{7^{2}}$

• Label: 1.49.ab
• Base field: $\F_{7^{2}}$
• 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:	$\F_{7^{2}}$
Dimension:	$1$
L-polynomial:	$1 - x + 49 x^{2}$
Frobenius angles:	$\pm0.477244201344$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-195}) \)
Galois group:	$C_2$
Jacobians:	$4$
Isomorphism classes:	4
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$49$	$2499$	$117796$	$5760195$	$282463489$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$49$	$2499$	$117796$	$5760195$	$282463489$	$13841501184$	$678223863121$	$33232920874755$	$1628413549492324$	$79792266724241379$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):

• $y^2=x^3+a^4 x+a^4$
• $y^2=x^3+a^{27} x+a^{28}$
• $y^2=x^3+a^{33} x+a^{34}$
• $y^2=x^3+a^{28} x+a^{28}$

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7^{2}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-195}) \).

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
1.49.b	$2$	(not in LMFDB)