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

• Label: 4.5.a_e_a_cc
• 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: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 + 2 x^{2} + 25 x^{4} )^{2}$
	$1 + 4 x^{2} + 54 x^{4} + 100 x^{6} + 625 x^{8}$
Frobenius angles:	$\pm0.282047108424$, $\pm0.282047108424$, $\pm0.717952891576$, $\pm0.717952891576$
Angle rank:	$1$ (numerical)

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$784$	$614656$	$239754256$	$203928109056$	$95470643144464$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$34$	$126$	$810$	$3126$	$15058$	$78126$	$387162$	$1953126$	$9786754$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=x^{10}+3 x^9+3 x^8+3 x^6+x^5+4 x^3+x+1$
• $y^2=x^{10}+4 x^9+2 x^8+x^7+4 x^6+4 x^5+3 x^4+4 x^3+x^2+4 x+2$
• $y^2=x^{10}+2 x^8+x^7+2 x^6+2 x^4+4 x^2+2 x+4$
• $y^2=x^9+2 x^8+4 x^7+3 x^6+3 x^4+x^3+2 x^2+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 2.5.a_c^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 $\F_{5^{2}}$ is 1.25.c^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-6}) \)$)$

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.am_cs_ake_bax	$3$	(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.g_t_bq_dg	$3$	(not in LMFDB)
4.5.m_cs_ke_bax	$3$	(not in LMFDB)