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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 4 x + 5 x^{2} )^{2}( 1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4} )$
	$1 - 12 x + 69 x^{2} - 252 x^{3} + 656 x^{4} - 1260 x^{5} + 1725 x^{6} - 1500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.147583617650$, $\pm0.185749715683$, $\pm0.480916950984$
Angle rank:	$1$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$52$	$317200$	$251539600$	$156234956800$	$102588049661812$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$20$	$126$	$640$	$3354$	$16562$	$79794$	$391680$	$1953126$	$9766100$

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 $\F_{5^{6}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae^2 $\times$ 2.5.ae_l 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.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 2.5.ae_l : \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ja^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$

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 $\times$ 2.25.g_l. The endomorphism algebra for each factor is:

  • 1.25.ag^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.25.g_l : \(\Q(\zeta_{12})\).
• Endomorphism algebra over $\F_{5^{3}}$
  The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae^2 $\times$ 1.125.e^2. The endomorphism algebra for each factor is:

  • 1.125.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.125.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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
4.5.ae_f_ae_q	$2$	(not in LMFDB)
4.5.ae_f_e_aq	$2$	(not in LMFDB)
4.5.e_f_ae_aq	$2$	(not in LMFDB)
4.5.e_f_e_q	$2$	(not in LMFDB)
4.5.m_cr_js_zg	$2$	(not in LMFDB)
4.5.a_am_a_di	$3$	(not in LMFDB)
4.5.a_g_a_l	$3$	(not in LMFDB)
4.5.m_cr_js_zg	$3$	(not in LMFDB)