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

• Label: 1.49.ao
• Base field: $\F_{7^{2}}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7^{2}}$
Dimension:	$1$
L-polynomial:	$( 1 - 7 x )^{2}$
	$1 - 14 x + 49 x^{2}$
Frobenius angles:	$0$, $0$
Angle rank:	$0$ (numerical)
Number field:	\(\Q\)
Galois group:	Trivial
Jacobians:	$1$
Isomorphism classes:	1
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$2304$	$116964$	$5760000$	$282441636$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$2304$	$116964$	$5760000$	$282441636$	$13841051904$	$678221425764$	$33232919040000$	$1628413517203236$	$79792265732661504$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $y^2=x^3+a^2 x$

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 the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.

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.o	$2$	(not in LMFDB)
1.49.a	$4$	(not in LMFDB)