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

• Label: 4.5.al_cj_aiq_wq
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - x + 5 x^{2} )( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$
	$1 - 11 x + 61 x^{2} - 224 x^{3} + 588 x^{4} - 1120 x^{5} + 1525 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0512862249088$, $\pm0.147583617650$, $\pm0.384619558242$, $\pm0.428216853436$
Angle rank:	$3$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$70$	$387100$	$264466720$	$138352636800$	$89100491984350$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$27$	$136$	$563$	$2915$	$15630$	$79319$	$392835$	$1952056$	$9756867$

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 $\times$ 1.5.ab $\times$ 2.5.ag_r 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 : \(\Q(\sqrt{-1}) \).
• 1.5.ab : \(\Q(\sqrt{-19}) \).
• 2.5.ag_r : \(\Q(\sqrt{2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm^2 $\times$ 1.15625.cc $\times$ 1.15625.ja. The endomorphism algebra for each factor is:

• 1.15625.afm^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
• 1.15625.cc : \(\Q(\sqrt{-19}) \).
• 1.15625.ja : \(\Q(\sqrt{-1}) \).

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 $\times$ 1.25.j $\times$ 2.25.ac_av. The endomorphism algebra for each factor is:

  • 1.25.ag : \(\Q(\sqrt{-1}) \).
  • 1.25.j : \(\Q(\sqrt{-19}) \).
  • 2.25.ac_av : \(\Q(\sqrt{2}, \sqrt{-3})\).
• Endomorphism algebra over $\F_{5^{3}}$
  The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae $\times$ 1.125.o $\times$ 2.125.a_afm. The endomorphism algebra for each factor is:

  • 1.125.ae : \(\Q(\sqrt{-1}) \).
  • 1.125.o : \(\Q(\sqrt{-19}) \).
  • 2.125.a_afm : \(\Q(\sqrt{2}, \sqrt{-3})\).

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.aj_bp_afc_mu	$2$	(not in LMFDB)
4.5.ad_f_a_abc	$2$	(not in LMFDB)
4.5.ab_b_ae_am	$2$	(not in LMFDB)
4.5.b_b_e_am	$2$	(not in LMFDB)
4.5.d_f_a_abc	$2$	(not in LMFDB)
4.5.j_bp_fc_mu	$2$	(not in LMFDB)
4.5.l_cj_iq_wq	$2$	(not in LMFDB)
4.5.af_q_abj_da	$3$	(not in LMFDB)
4.5.b_b_e_am	$3$	(not in LMFDB)