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

• Label: 4.5.c_c_i_bf
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$1 + 2 x + 2 x^{2} + 8 x^{3} + 31 x^{4} + 40 x^{5} + 50 x^{6} + 250 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.213073691352$, $\pm0.392183190680$, $\pm0.713073691352$, $\pm0.892183190680$
Angle rank:	$2$ (numerical)
Number field:	8.0.2136658176.4
Galois group:	$D_4\times C_2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1009$	$412681$	$286200832$	$170305607761$	$94215066004849$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$26$	$146$	$694$	$3088$	$15626$	$78688$	$391398$	$1946666$	$9765626$

Jacobians and polarizations

This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 8.0.2136658176.4.

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

The base change of $A$ to $\F_{5^{4}}$ is 2.625.bi_bdr^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.46224.1$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 4.25.a_bi_a_bdr and its endomorphism algebra is 8.0.2136658176.4.

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.ac_c_ai_bf	$2$	(not in LMFDB)
4.5.ae_f_i_abs	$3$	(not in LMFDB)
4.5.c_f_o_q	$3$	(not in LMFDB)