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

• Label: 4.5.an_dd_ame_bgm
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$
	$1 - 13 x + 81 x^{2} - 316 x^{3} + 844 x^{4} - 1580 x^{5} + 2025 x^{6} - 1625 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0512862249088$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.384619558242$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$42$	$298620$	$272022912$	$156954672000$	$93754335863202$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$19$	$140$	$643$	$3073$	$15502$	$78253$	$391395$	$1953644$	$9763819$

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.ad $\times$ 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.ad : \(\Q(\sqrt{-11}) \).
• 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.acw $\times$ 1.15625.ja. The endomorphism algebra for each factor is:

• 1.15625.afm^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
• 1.15625.acw : \(\Q(\sqrt{-11}) \).
• 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.b $\times$ 2.25.ac_av. The endomorphism algebra for each factor is:

  • 1.25.ag : \(\Q(\sqrt{-1}) \).
  • 1.25.b : \(\Q(\sqrt{-11}) \).
  • 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.s $\times$ 2.125.a_afm. The endomorphism algebra for each factor is:

  • 1.125.ae : \(\Q(\sqrt{-1}) \).
  • 1.125.s : \(\Q(\sqrt{-11}) \).
  • 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.ah_v_abo_cy	$2$	(not in LMFDB)
4.5.af_j_e_abs	$2$	(not in LMFDB)
4.5.ab_ad_i_e	$2$	(not in LMFDB)
4.5.b_ad_ai_e	$2$	(not in LMFDB)
4.5.f_j_ae_abs	$2$	(not in LMFDB)
4.5.h_v_bo_cy	$2$	(not in LMFDB)
4.5.n_dd_me_bgm	$2$	(not in LMFDB)
4.5.ah_y_abx_dq	$3$	(not in LMFDB)
4.5.ab_ad_i_e	$3$	(not in LMFDB)