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

• Label: 4.5.a_a_a_by
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 + 25 x^{4} )^{2}$
	$1 + 50 x^{4} + 625 x^{8}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.250000000000$, $\pm0.750000000000$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2, 13$

This isogeny class is not 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:	$[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$676$	$456976$	$244171876$	$208827064576$	$95367451171876$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$26$	$126$	$826$	$3126$	$15626$	$78126$	$385626$	$1953126$	$9765626$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=x^{10}+x^9+4 x^8+2 x^7+3 x^6+4 x^4+2 x^3+3 x^2+4 x+3$
• $y^2=x^{10}+2 x^8+2 x^6+3 x^4+3 x^2+4$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 2.5.a_a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(i, \sqrt{10})\)$)$

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

The base change of $A$ to $\F_{5^{4}}$ is 1.625.by^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_by^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_az	$3$	(not in LMFDB)
4.5.a_au_a_fu	$8$	(not in LMFDB)
4.5.a_ak_a_by	$8$	(not in LMFDB)