Abelian variety isogeny class 4.5.a_a_a_az over $\F_{5}$

• Label: 4.5.a_a_a_az
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$1 - 25 x^{4} + 625 x^{8}$
Frobenius angles:	$\pm0.0833333333333$, $\pm0.416666666667$, $\pm0.583333333333$, $\pm0.916666666667$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(i, \sqrt{3}, \sqrt{10})\)
Galois group:	$C_2^3$
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$601$	$361201$	$244171876$	$130466162401$	$95367421875001$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$26$	$126$	$526$	$3126$	$15626$	$78126$	$393126$	$1953126$	$9765626$

Jacobians and polarizations

It is unknown whether this isogeny class contains a Jacobian or whether it is principally polarizable.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{3}, \sqrt{10})\).

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

The base change of $A$ to $\F_{5^{12}}$ is 1.244140625.bufy^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$

  The base change of $A$ to $\F_{5^{2}}$ is 2.25.a_az^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{12})\)$)$

• Endomorphism algebra over $\F_{5^{3}}$

  The base change of $A$ to $\F_{5^{3}}$ is 2.125.a_a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(i, \sqrt{10})\)$)$

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

  The base change of $A$ to $\F_{5^{4}}$ is 1.625.az^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{5^{6}}$

  The base change of $A$ to $\F_{5^{6}}$ is 2.15625.a_bufy^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q(\sqrt{-1}) \) with the following ramification data at primes above $5$, and unramified at all archimedean places:

  $v$	($ 5 $,\( \pi + 2 \))	($ 5 $,\( \pi + 3 \))
  $\operatorname{inv}_v$	$1/2$	$1/2$

  where $\pi$ is a root of $x^{2} + 1$.

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
4.5.a_a_a_by	$3$	(not in LMFDB)
4.5.a_ak_a_cx	$8$	(not in LMFDB)
4.5.a_a_a_z	$8$	(not in LMFDB)