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

• Label: 4.5.g_j_abe_afg
• 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:	$( 1 + 4 x + 5 x^{2} )^{2}( 1 - 2 x - x^{2} - 10 x^{3} + 25 x^{4} )$
	$1 + 6 x + 9 x^{2} - 30 x^{3} - 136 x^{4} - 150 x^{5} + 225 x^{6} + 750 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0190830490162$, $\pm0.685749715683$, $\pm0.852416382350$, $\pm0.852416382350$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1300$	$192400$	$182790400$	$156234956800$	$88493687132500$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$8$	$90$	$640$	$2892$	$15626$	$76956$	$391680$	$1945458$	$9767048$

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 $\F_{5^{12}}$.

Endomorphism algebra over $\F_{5}$

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

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

The base change of $A$ to $\F_{5^{12}}$ is 1.244140625.abiuc^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.ag_l. The endomorphism algebra for each factor is:

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

  • 1.125.aw^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.125.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• Endomorphism algebra over $\F_{5^{4}}$
  The base change of $A$ to $\F_{5^{4}}$ is 1.625.o^2 $\times$ 2.625.ao_aqn. The endomorphism algebra for each factor is:

  • 1.625.o^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.625.ao_aqn : \(\Q(\zeta_{12})\).
• Endomorphism algebra over $\F_{5^{6}}$
  The base change of $A$ to $\F_{5^{6}}$ is 1.15625.aja^2 $\times$ 1.15625.ja^2. The endomorphism algebra for each factor is:

  • 1.15625.aja^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.15625.ja^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.ak_bp_adq_hc	$2$	(not in LMFDB)
4.5.ag_j_be_afg	$2$	(not in LMFDB)
4.5.ac_ah_c_ce	$2$	(not in LMFDB)
4.5.c_ah_ac_ce	$2$	(not in LMFDB)
4.5.k_bp_dq_hc	$2$	(not in LMFDB)
4.5.ag_s_abw_ep	$3$	(not in LMFDB)
4.5.a_j_am_bs	$3$	(not in LMFDB)
4.5.m_cu_kq_bcg	$3$	(not in LMFDB)