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

• Label: 2.47.ad_abm
• 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:	$1 - 3 x - 38 x^{2} - 141 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.0964532585651$, $\pm0.763119925232$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-179})\)
Galois group:	$C_2^2$
Jacobians:	$35$
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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2028$	$4696848$	$10697351184$	$23824986928704$	$52592932853666868$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$45$	$2125$	$103032$	$4882489$	$229317975$	$10779316990$	$506624543505$	$23811284781649$	$1119130595587944$	$52599132507923125$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 35 curves (of which all are hyperelliptic):

• $y^2=44 x^6+22 x^5+5 x^3+32 x^2+x+41$
• $y^2=23 x^6+35 x^5+30 x^4+25 x^3+6 x^2+40 x$
• $y^2=38 x^6+34 x^5+19 x^4+27 x^2+15 x+33$
• $y^2=42 x^6+18 x^5+36 x^4+26 x^3+12 x^2+46 x+42$
• $y^2=15 x^6+11 x^5+39 x^4+29 x^3+6 x^2+24 x+24$
• $y^2=19 x^6+33 x^5+23 x^4+5 x^3+28 x^2+21 x+46$
• $y^2=24 x^6+42 x^5+21 x^4+22 x^3+32 x^2+13 x+46$
• $y^2=4 x^6+40 x^5+27 x^4+37 x^3+26 x^2+6 x+30$
• $y^2=20 x^6+35 x^5+25 x^4+6 x^3+9 x^2+33 x+20$
• $y^2=34 x^6+33 x^5+46 x^4+16 x^3+10 x^2+15 x+34$
• $y^2=5 x^6+27 x^5+33 x^4+40 x^3+26 x^2+36 x+10$
• $y^2=7 x^6+7 x^5+33 x^4+7 x^3+9 x^2+35 x+7$
• $y^2=11 x^6+36 x^5+22 x^4+21 x^3+7 x^2+30 x+11$
• $y^2=29 x^6+20 x^5+10 x^4+34 x^3+21 x^2+10 x+45$
• $y^2=6 x^6+27 x^5+24 x^4+46 x^3+28 x^2+37 x+3$
• $y^2=2 x^6+43 x^5+8 x^4+43 x^3+5 x+12$
• $y^2=42 x^5+23 x^4+39 x^3+10 x^2+28 x+1$
• $y^2=17 x^5+20 x^4+20 x^3+37 x^2+19 x+45$
• $y^2=17 x^6+6 x^5+5 x^4+32 x^3+46 x^2+44 x+8$
• $y^2=35 x^6+3 x^5+29 x^4+8 x^3+16 x^2+45 x+5$
• and

• $y^2=12 x^6+6 x^5+10 x^4+17 x^3+16 x^2+3 x+12$
• $y^2=27 x^6+5 x^5+11 x^4+13 x^3+27 x^2+17 x+44$
• $y^2=17 x^6+2 x^5+35 x^4+42 x^3+45 x^2+6 x+17$
• $y^2=8 x^6+36 x^5+12 x^4+34 x^3+29 x^2+39 x+25$
• $y^2=14 x^6+15 x^5+2 x^4+22 x^3+43 x^2+22 x+14$
• $y^2=5 x^6+44 x^5+24 x^4+13 x^3+10 x^2+5 x+33$
• $y^2=37 x^6+32 x^5+43 x^4+5 x^3+16 x^2+25 x+13$
• $y^2=14 x^6+17 x^5+38 x^4+46 x^2+30 x+43$
• $y^2=22 x^6+18 x^5+35 x^4+10 x^3+8 x^2+6 x+12$
• $y^2=x^6+3 x^5+32 x^4+40 x^3+11 x^2+5 x+9$
• $y^2=40 x^6+9 x^5+2 x^4+46 x^3+37 x^2+32 x+9$
• $y^2=45 x^6+15 x^5+37 x^4+27 x^3+24 x^2+46 x+4$
• $y^2=33 x^6+20 x^5+27 x^4+36 x^3+27 x^2+9 x+33$
• $y^2=29 x^6+21 x^5+23 x^4+15 x^3+11 x^2+26 x+33$
• $y^2=13 x^6+7 x^5+5 x^4+11 x^3+19 x^2+36 x+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{3}}$.

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-179})\).

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

The base change of $A$ to $\F_{47^{3}}$ is 1.103823.apg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-179}) \)$)$

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
2.47.d_abm	$2$	(not in LMFDB)
2.47.g_dz	$3$	(not in LMFDB)
2.47.ag_dz	$6$	(not in LMFDB)