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

• Label: 4.5.am_cu_akp_bcc
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4} )$
	$1 - 12 x + 72 x^{2} - 275 x^{3} + 730 x^{4} - 1375 x^{5} + 1800 x^{6} - 1500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.200000000000$, $\pm0.265942140215$, $\pm0.400000000000$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$66$	$421740$	$336444768$	$175781232000$	$99467228373216$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$26$	$165$	$714$	$3259$	$15911$	$78800$	$391154$	$1950519$	$9756601$

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 $\times$ 1.5.ad $\times$ 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 : \(\Q(\sqrt{-1}) \).
• 1.5.ad : \(\Q(\sqrt{-11}) \).
• 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 $\times$ 1.9765625.ell. 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 : \(\Q(\sqrt{-1}) \).
• 1.9765625.ell : \(\Q(\sqrt{-11}) \).

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.f_z. The endomorphism algebra for each factor is:

  • 1.25.ag : \(\Q(\sqrt{-1}) \).
  • 1.25.b : \(\Q(\sqrt{-11}) \).
  • 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.cf $\times$ 1.3125.cy $\times$ 2.3125.a_ajgk. The endomorphism algebra for each factor is:

  • 1.3125.cf : \(\Q(\sqrt{-11}) \).
  • 1.3125.cy : \(\Q(\sqrt{-1}) \).
  • 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.ag_s_abj_cs	$2$	(not in LMFDB)
4.5.ae_i_f_abe	$2$	(not in LMFDB)
4.5.ac_c_af_be	$2$	(not in LMFDB)
4.5.c_c_f_be	$2$	(not in LMFDB)
4.5.e_i_af_abe	$2$	(not in LMFDB)
4.5.g_s_bj_cs	$2$	(not in LMFDB)
4.5.m_cu_kp_bcc	$2$	(not in LMFDB)