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

• Label: 2.47.ak_cb
• 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:	$1 - 10 x + 53 x^{2} - 470 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.0934990704185$, $\pm0.573167596248$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-22})\)
Galois group:	$C_2^2$
Jacobians:	$36$
Isomorphism classes:	38
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1783$	$4890769$	$10694455396$	$23789918217481$	$52604763029133943$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$38$	$2216$	$103004$	$4875300$	$229369558$	$10779294422$	$506622362074$	$23811296104324$	$1119130590665348$	$52599132379842536$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=42 x^6+30 x^5+25 x^4+7 x^3+34 x^2+15 x+17$
• $y^2=45 x^6+9 x^5+18 x^3+4 x^2+41 x+44$
• $y^2=40 x^6+16 x^5+43 x^4+18 x^3+9 x^2+36 x+23$
• $y^2=44 x^6+19 x^4+4 x^3+11 x^2+20 x+41$
• $y^2=11 x^6+32 x^5+7 x^4+18 x^3+35 x^2+42 x+37$
• $y^2=41 x^6+5 x^5+16 x^4+6 x^3+39 x^2+14 x+42$
• $y^2=14 x^6+3 x^5+38 x^4+44 x^3+12 x^2+3 x+37$
• $y^2=13 x^6+12 x^5+43 x^3+x^2+45 x+17$
• $y^2=15 x^6+25 x^5+16 x^4+24 x^3+37 x^2+8 x+29$
• $y^2=29 x^6+6 x^5+16 x^4+32 x^3+9 x^2+16 x+37$
• $y^2=10 x^6+26 x^5+9 x^4+42 x^3+12 x^2+23 x+11$
• $y^2=3 x^6+40 x^5+9 x^4+18 x^3+8 x^2+30 x+30$
• $y^2=4 x^6+25 x^5+x^4+4 x^3+46 x^2+3 x+3$
• $y^2=x^6+40 x^5+9 x^4+13 x^3+27 x^2+30 x+23$
• $y^2=41 x^6+36 x^5+43 x^4+43 x^3+7 x^2+4 x+23$
• $y^2=25 x^6+43 x^5+37 x^4+4 x^3+15 x^2+28$
• $y^2=38 x^6+31 x^5+25 x^4+13 x^3+26 x^2+38 x+8$
• $y^2=29 x^6+45 x^5+33 x^4+4 x^3+38 x^2+24 x+41$
• $y^2=40 x^6+27 x^5+16 x^4+40 x^3+27 x^2+16 x+25$
• $y^2=13 x^6+46 x^4+x^3+19 x^2+22 x+5$
• and

• $y^2=33 x^6+24 x^5+43 x^3+x^2+24 x+43$
• $y^2=12 x^6+31 x^5+43 x^4+45 x^3+11 x^2+27 x+35$
• $y^2=31 x^6+11 x^5+24 x^4+16 x^3+25 x^2+8 x+36$
• $y^2=26 x^6+28 x^5+21 x^4+35 x^3+5 x^2+34 x+13$
• $y^2=45 x^6+35 x^5+6 x^4+17 x^3+14 x^2+34 x+25$
• $y^2=21 x^6+37 x^5+41 x^4+5 x^3+39 x^2+15 x+14$
• $y^2=22 x^6+42 x^5+18 x^4+17 x^3+16 x^2+41 x+30$
• $y^2=34 x^6+45 x^5+45 x^4+36 x^3+12 x+44$
• $y^2=8 x^6+33 x^5+13 x^4+4 x^3+35 x^2+44 x+35$
• $y^2=14 x^6+6 x^5+21 x^4+33 x^2+40 x+30$
• $y^2=30 x^6+29 x^5+6 x^4+21 x^3+7 x^2+5 x+13$
• $y^2=15 x^6+7 x^5+8 x^4+32 x^3+19 x^2+21 x+5$
• $y^2=29 x^6+15 x^5+31 x^4+37 x^3+5 x^2+40 x+13$
• $y^2=17 x^6+38 x^5+17 x^4+9 x^3+23 x^2+35 x+3$
• $y^2=11 x^6+13 x^5+40 x^4+11 x^3+7 x^2+6 x+30$
• $y^2=25 x^6+2 x^5+19 x^4+12 x^3+46 x^2+42 x+35$

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 \(\Q(\sqrt{-3}, \sqrt{-22})\).

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

The base change of $A$ to $\F_{47^{3}}$ is 1.103823.apu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$

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.k_cb	$2$	(not in LMFDB)
2.47.u_hm	$3$	(not in LMFDB)
2.47.au_hm	$6$	(not in LMFDB)