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

• Label: 3.5.b_p_k
• Base field: $\F_{5}$
• Dimension: $3$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$( 1 + x + 5 x^{2} )( 1 + 5 x^{2} )^{2}$
	$1 + x + 15 x^{2} + 10 x^{3} + 75 x^{4} + 25 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.5$, $\pm0.5$, $\pm0.571783146564$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$252$	$45360$	$1778112$	$197406720$	$31533843852$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$55$	$112$	$495$	$3227$	$16180$	$77567$	$388415$	$1955632$	$9774175$

Jacobians and polarizations

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

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^2 $\times$ 1.5.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$
• 1.5.b : \(\Q(\sqrt{-19}) \).

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

The base change of $A$ to $\F_{5^{2}}$ is 1.25.j $\times$ 1.25.k^2. The endomorphism algebra for each factor is:

• 1.25.j : \(\Q(\sqrt{-19}) \).
• 1.25.k^2 : $\mathrm{M}_{2}(B)$, 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.ab_p_ak	$2$	3.25.bd_nr_dmk
3.5.b_a_af	$3$	(not in LMFDB)
3.5.ab_af_k	$4$	(not in LMFDB)