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

• Label: 2.47.k_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:	$\pm0.426832403752$, $\pm0.906500929582$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-22})\)
Galois group:	$C_2^2$
Jacobians:	$36$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2743$	$4890769$	$10864726756$	$23789918217481$	$52593502189241143$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$58$	$2216$	$104644$	$4875300$	$229320458$	$10779294422$	$506623878854$	$23811296104324$	$1119130355540188$	$52599132379842536$

Jacobians and polarizations

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

• $y^2=22 x^6+9 x^5+31 x^4+35 x^3+29 x^2+28 x+38$
• $y^2=37 x^6+45 x^5+43 x^3+20 x^2+17 x+32$
• $y^2=12 x^6+33 x^5+27 x^4+43 x^3+45 x^2+39 x+21$
• $y^2=37 x^6+32 x^4+29 x^3+21 x^2+4 x+27$
• $y^2=21 x^6+44 x^5+39 x^4+13 x^3+7 x^2+46 x+45$
• $y^2=17 x^6+25 x^5+33 x^4+30 x^3+7 x^2+23 x+22$
• $y^2=31 x^6+10 x^5+17 x^4+37 x^3+40 x^2+10 x+45$
• $y^2=12 x^6+40 x^5+18 x^3+19 x^2+9 x+41$
• $y^2=3 x^6+5 x^5+22 x^4+33 x^3+45 x^2+11 x+34$
• $y^2=34 x^6+20 x^5+22 x^4+44 x^3+30 x^2+22 x+45$
• $y^2=2 x^6+24 x^5+30 x^4+46 x^3+40 x^2+14 x+21$
• $y^2=15 x^6+12 x^5+45 x^4+43 x^3+40 x^2+9 x+9$
• $y^2=20 x^6+31 x^5+5 x^4+20 x^3+42 x^2+15 x+15$
• $y^2=5 x^6+12 x^5+45 x^4+18 x^3+41 x^2+9 x+21$
• $y^2=27 x^6+26 x^5+18 x^4+18 x^3+39 x^2+29 x+14$
• $y^2=31 x^6+27 x^5+44 x^4+20 x^3+28 x^2+46$
• $y^2=17 x^6+25 x^5+5 x^4+12 x^3+24 x^2+17 x+11$
• $y^2=34 x^6+9 x^5+16 x^4+29 x^3+17 x^2+33 x+27$
• $y^2=8 x^6+43 x^5+22 x^4+8 x^3+43 x^2+22 x+5$
• $y^2=12 x^6+28 x^4+19 x^3+32 x^2+42 x+1$
• and

• $y^2=16 x^6+33 x^5+18 x^3+19 x^2+33 x+18$
• $y^2=13 x^6+14 x^5+27 x^4+37 x^3+8 x^2+41 x+34$
• $y^2=25 x^6+21 x^5+33 x^4+22 x^3+5 x^2+11 x+26$
• $y^2=24 x^6+15 x^5+23 x^4+7 x^3+x^2+35 x+12$
• $y^2=37 x^6+34 x^5+30 x^4+38 x^3+23 x^2+29 x+31$
• $y^2=11 x^6+44 x^5+17 x^4+25 x^3+7 x^2+28 x+23$
• $y^2=16 x^6+22 x^5+43 x^4+38 x^3+33 x^2+17 x+9$
• $y^2=35 x^6+9 x^5+9 x^4+26 x^3+40 x+37$
• $y^2=11 x^6+16 x^5+12 x^4+29 x^3+7 x^2+37 x+7$
• $y^2=23 x^6+30 x^5+11 x^4+24 x^2+12 x+9$
• $y^2=9 x^6+4 x^5+30 x^4+11 x^3+35 x^2+25 x+18$
• $y^2=28 x^6+35 x^5+40 x^4+19 x^3+x^2+11 x+25$
• $y^2=4 x^6+28 x^5+14 x^4+44 x^3+25 x^2+12 x+18$
• $y^2=41 x^6+17 x^5+41 x^4+30 x^3+14 x^2+7 x+10$
• $y^2=8 x^6+18 x^5+12 x^4+8 x^3+35 x^2+30 x+9$
• $y^2=31 x^6+10 x^5+x^4+13 x^3+42 x^2+22 x+34$

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.pu^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.ak_cb	$2$	(not in LMFDB)
2.47.au_hm	$3$	(not in LMFDB)
2.47.a_ag	$6$	(not in LMFDB)