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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$( 1 + 2 x + 5 x^{2} )( 1 + 4 x + 5 x^{2} )^{2}$
	$1 + 10 x + 47 x^{2} + 132 x^{3} + 235 x^{4} + 250 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.647583617650$, $\pm0.852416382350$, $\pm0.852416382350$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$800$	$12800$	$1757600$	$262144000$	$29842420000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$20$	$112$	$668$	$3056$	$15860$	$77072$	$393788$	$1947856$	$9766100$

Jacobians and polarizations

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

• $4 x^4+x^3 z+4 x^2 z^2+x z^3+y^4+y^2 z^2=0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.c $\times$ 1.5.e^2 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.c : \(\Q(\sqrt{-1}) \).
• 1.5.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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

The base change of $A$ to $\F_{5^{4}}$ is 1.625.o^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$
  The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag^2 $\times$ 1.25.g. The endomorphism algebra for each factor is:

  • 1.25.ag^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.25.g : \(\Q(\sqrt{-1}) \).

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
3.5.ak_bv_afc	$2$	3.25.ag_bn_adg
3.5.ag_p_abc	$2$	3.25.ag_bn_adg
3.5.ac_ab_m	$2$	3.25.ag_bn_adg
3.5.c_ab_am	$2$	3.25.ag_bn_adg
3.5.g_p_bc	$2$	3.25.ag_bn_adg
3.5.ac_i_as	$3$	(not in LMFDB)