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

• Label: 4.5.d_ab_ad_q
• 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:	$1 + 3 x - x^{2} - 3 x^{3} + 16 x^{4} - 15 x^{5} - 25 x^{6} + 375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.144130581889$, $\pm0.417838834988$, $\pm0.817838834988$, $\pm0.944130581889$
Angle rank:	$2$ (numerical)
Number field:	8.0.1816890625.4
Galois group:	$C_2^2:C_4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$976$	$249856$	$306445456$	$148334510080$	$93154617968896$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$15$	$153$	$607$	$3054$	$15975$	$79053$	$392287$	$1953909$	$9757330$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 8.0.1816890625.4.

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

The base change of $A$ to $\F_{5^{5}}$ is 2.3125.abk_accw^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.8525.1$)$

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.ad_ab_d_q	$2$	(not in LMFDB)
4.5.ac_ab_ai_bp	$5$	(not in LMFDB)
4.5.ac_t_abc_fl	$5$	(not in LMFDB)