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

• Label: 4.5.an_dd_amd_bgi
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $2$
• Ordinary: no
• 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:	$( 1 - 4 x + 5 x^{2} )^{2}( 1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4} )$
	$1 - 13 x + 81 x^{2} - 315 x^{3} + 840 x^{4} - 1575 x^{5} + 2025 x^{6} - 1625 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.147583617650$, $\pm0.200000000000$, $\pm0.400000000000$
Angle rank:	$1$ (numerical)

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

Newton polygon

$p$-rank:	$2$
Slopes:	$[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$44$	$312400$	$285043484$	$166666649600$	$100060969290304$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$19$	$143$	$679$	$3278$	$16219$	$79863$	$393359$	$1952873$	$9754074$

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 $\F_{5^{10}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae^2 $\times$ 2.5.af_p 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.af_p : \(\Q(\zeta_{5})\).

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

The base change of $A$ to $\F_{5^{10}}$ is 1.9765625.ajgk^2 $\times$ 1.9765625.sg^2. The endomorphism algebra for each factor is:

• 1.9765625.ajgk^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
• 1.9765625.sg^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.f_z. The endomorphism algebra for each factor is:

  • 1.25.ag^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.25.f_z : \(\Q(\zeta_{5})\).
• Endomorphism algebra over $\F_{5^{5}}$
  The base change of $A$ to $\F_{5^{5}}$ is 1.3125.cy^2 $\times$ 2.3125.a_ajgk. The endomorphism algebra for each factor is:

  • 1.3125.cy^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.3125.a_ajgk : the quaternion algebra over \(\Q(\sqrt{5}) \) ramified at both real infinite places.

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.af_j_f_abo	$2$	(not in LMFDB)
4.5.ad_b_af_bo	$2$	(not in LMFDB)
4.5.d_b_f_bo	$2$	(not in LMFDB)
4.5.f_j_af_abo	$2$	(not in LMFDB)
4.5.n_dd_md_bgi	$2$	(not in LMFDB)
4.5.ab_g_a_p	$3$	(not in LMFDB)