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

• Label: 4.5.d_l_y_cc
• 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:	$1 + 3 x + 11 x^{2} + 24 x^{3} + 54 x^{4} + 120 x^{5} + 275 x^{6} + 375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.276642081063$, $\pm0.524248544986$, $\pm0.609975414396$, $\pm0.857581878319$
Angle rank:	$2$ (numerical)
Number field:	8.0.82653950016.2
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 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1488$	$660672$	$245769984$	$152887428864$	$99219782296848$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$39$	$126$	$627$	$3249$	$15834$	$76365$	$390915$	$1953126$	$9770319$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{6}}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 8.0.82653950016.2.

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

The base change of $A$ to $\F_{5^{6}}$ is 2.15625.ea_ghq^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.287496.1$)$

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.n_dh_ra_dgw and its endomorphism algebra is 8.0.82653950016.2.

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

  The base change of $A$ to $\F_{5^{3}}$ is the simple isogeny class 4.125.a_ea_a_ghq and its endomorphism algebra is 8.0.82653950016.2.

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_l_ay_cc	$3$	(not in LMFDB)
4.5.a_an_a_dg	$3$	(not in LMFDB)
4.5.a_an_a_dg	$6$	(not in LMFDB)