Abelian variety isogeny class 3.5.a_p_a over $\F_{5}$

• Label: 3.5.a_p_a
• Base field: $\F_{5}$
• Dimension: $3$
• $p$-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:	$3$
L-polynomial:	$( 1 + 5 x^{2} )^{3}$
	$1 + 15 x^{2} + 75 x^{4} + 125 x^{6}$
Frobenius angles:	$\pm0.5$, $\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$1$

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:	$[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$216$	$46656$	$2000376$	$191102976$	$30546884376$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$56$	$126$	$476$	$3126$	$16376$	$78126$	$386876$	$1953126$	$9784376$

Jacobians and polarizations

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

• $2x^4+3x^2yz+xy^3+xz^3+y^2z^2=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.a^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-5}) \)$)$

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

The base change of $A$ to $\F_{5^{2}}$ is 1.25.k^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ 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
3.5.a_a_a	$3$	(not in LMFDB)
3.5.a_af_a	$4$	(not in LMFDB)
3.5.a_f_a	$8$	(not in LMFDB)