Abelian variety isogeny class 2.47.ao_du over $\F_{47}$

• Label: 2.47.ao_du
• Base field: $\F_{47}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$1 - 14 x + 98 x^{2} - 658 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.00677459636375$, $\pm0.493225403636$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$6$

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1636$	$4875280$	$10716894484$	$23768355078400$	$52591487804911156$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$2210$	$103222$	$4870878$	$229311674$	$10779215330$	$506621923982$	$23811267425278$	$1119130394682034$	$52599132235830050$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=37 x^6+31 x^5+27 x^4+4 x^3+36 x^2+6 x+7$
• $y^2=33 x^6+29 x^5+31 x^4+31 x^2+18 x+33$
• $y^2=10 x^6+22 x^5+37 x^4+37 x^2+25 x+10$
• $y^2=44 x^6+37 x^5+15 x^4+23 x^3+22 x^2+31$
• $y^2=22 x^6+3 x^5+42 x^4+7 x^3+4 x+38$
• $y^2=25 x^6+32 x^5+9 x^4+9 x^2+15 x+25$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{4}}$.

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{5})\).

Endomorphism algebra over $\overline{\F}_{47}$

The base change of $A$ to $\F_{47^{4}}$ is 1.4879681.agni^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{47^{2}}$

  The base change of $A$ to $\F_{47^{2}}$ is the simple isogeny class 2.2209.a_agni and its endomorphism algebra is \(\Q(i, \sqrt{5})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.47.o_du	$2$	(not in LMFDB)
2.47.a_ae	$8$	(not in LMFDB)
2.47.a_e	$8$	(not in LMFDB)