Abelian variety isogeny class 1.625.az over $\F_{5^{4}}$

• Label: 1.625.az
• Base field: $\F_{5^{4}}$
• 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:	$\F_{5^{4}}$
Dimension:	$1$
L-polynomial:	$1 - 25 x + 625 x^{2}$
Frobenius angles:	$\pm0.333333333333$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$601$	$391251$	$244171876$	$152588281251$	$95367421875001$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$601$	$391251$	$244171876$	$152588281251$	$95367421875001$	$59604644287109376$	$37252902978515625001$	$23283064365539550781251$	$14551915228374481201171876$	$9094947017729377746582031251$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{12}}$.

Endomorphism algebra over $\F_{5^{4}}$

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

Endomorphism algebra over $\overline{\F}_{5^{4}}$

The base change of $A$ to $\F_{5^{12}}$ is the simple isogeny class 1.244140625.bufy and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

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
1.625.z	$2$	(not in LMFDB)
1.625.by	$3$	(not in LMFDB)
1.625.aby	$6$	(not in LMFDB)