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

• Label: 4.5.am_cs_ake_bax
• 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:	$( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )^{2}$
	$1 - 12 x + 70 x^{2} - 264 x^{3} + 699 x^{4} - 1320 x^{5} + 1750 x^{6} - 1500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0512862249088$, $\pm0.0512862249088$, $\pm0.384619558242$, $\pm0.384619558242$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$49$	$305809$	$239754256$	$132002149041$	$84618786129409$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$22$	$126$	$534$	$2754$	$15058$	$78282$	$392358$	$1953126$	$9755062$

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 2.5.ag_r^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{2}, \sqrt{-3})\)$)$

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

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-6}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$

  The base change of $A$ to $\F_{5^{2}}$ is 2.25.ac_av^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{2}, \sqrt{-3})\)$)$

• Endomorphism algebra over $\F_{5^{3}}$

  The base change of $A$ to $\F_{5^{3}}$ is 2.125.a_afm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{2}, \sqrt{-3})\)$)$

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.a_ac_a_av	$2$	(not in LMFDB)
4.5.m_cs_ke_bax	$2$	(not in LMFDB)
4.5.ag_t_abq_dg	$3$	(not in LMFDB)
4.5.a_ac_a_av	$3$	(not in LMFDB)
4.5.a_e_a_cc	$3$	(not in LMFDB)
4.5.g_t_bq_dg	$3$	(not in LMFDB)
4.5.m_cs_ke_bax	$3$	(not in LMFDB)