Abelian variety isogeny class 2.53.ac_abx over $\F_{53}$

• Label: 2.53.ac_abx
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 - 2 x - 49 x^{2} - 106 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.122804766083$, $\pm0.789471432750$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-13})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2653$	$7611457$	$22072450624$	$62297469273529$	$174876596704186573$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$2708$	$148258$	$7895268$	$418169492$	$22164764438$	$1174712921252$	$62259697536196$	$3299763809131354$	$174887470205226068$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{3}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-13})\).

Endomorphism algebra over $\overline{\F}_{53}$

The base change of $A$ to $\F_{53^{3}}$ is 1.148877.aly^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

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
2.53.c_abx	$2$	(not in LMFDB)
2.53.e_eg	$3$	(not in LMFDB)
2.53.ae_eg	$6$	(not in LMFDB)