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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 20 x^{5} + 150 x^{6} - 500 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0702545587912$, $\pm0.239421005470$, $\pm0.489421005470$, $\pm0.820254558791$
Angle rank:	$2$ (numerical)
Number field:	8.0.1698758656.6
Galois group:	$D_4\times C_2$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$256$	$335872$	$240034048$	$152211816448$	$92850465280256$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$22$	$122$	$626$	$3042$	$16102$	$77562$	$388698$	$1951970$	$9768182$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 7 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{8}}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 8.0.1698758656.6.

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

The base change of $A$ to $\F_{5^{8}}$ is 2.390625.ablc_bbvus^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-34 +16 \sqrt{2}})\)$)$

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.ae_i_fs_abqo and its endomorphism algebra is 8.0.1698758656.6.

• Endomorphism algebra over $\F_{5^{4}}$

  The base change of $A$ to $\F_{5^{4}}$ is the simple isogeny class 4.625.a_ablc_a_bbvus and its endomorphism algebra is 8.0.1698758656.6.

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.e_g_e_c	$2$	(not in LMFDB)
4.5.a_c_aq_c	$4$	(not in LMFDB)
4.5.a_c_q_c	$4$	(not in LMFDB)