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

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

Invariants

Base field:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$1 + 14 x + 138 x^{2} + 658 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.612950816214$, $\pm0.735257010414$
Angle rank:	$2$ (numerical)
Number field:	4.0.424400.1
Galois group:	$D_{4}$
Jacobians:	$42$
Isomorphism classes:	56
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3020$	$5061520$	$10667790620$	$23829231238400$	$52602282046325500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$2290$	$102746$	$4883358$	$229358742$	$10779048370$	$506623525106$	$23811286879998$	$1119130493728142$	$52599132019518450$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=41 x^6+25 x^5+33 x^4+22 x^3+27 x^2+20 x$
• $y^2=37 x^6+28 x^5+46 x^4+34 x^3+4 x^2+44 x+40$
• $y^2=6 x^6+7 x^5+41 x^4+14 x^2+34 x+1$
• $y^2=42 x^6+5 x^5+40 x^4+2 x^3+10 x^2+13 x+27$
• $y^2=44 x^6+30 x^5+31 x^4+33 x^3+3 x^2+21 x+13$
• $y^2=14 x^6+12 x^5+8 x^4+17 x^3+31 x^2+7 x+36$
• $y^2=3 x^6+12 x^5+21 x^4+31 x^3+6 x^2+13 x+16$
• $y^2=6 x^6+27 x^5+38 x^4+37 x^3+20 x^2+31 x+37$
• $y^2=40 x^6+20 x^5+40 x^4+20 x^3+46 x^2+41 x+23$
• $y^2=25 x^6+31 x^5+30 x^4+40 x^3+23 x^2+35 x+18$
• $y^2=10 x^5+14 x^4+8 x^3+13 x^2+15 x+34$
• $y^2=31 x^5+23 x^4+35 x^3+32 x^2+29 x+14$
• $y^2=4 x^6+7 x^5+10 x^4+18 x^2+19 x+36$
• $y^2=24 x^6+21 x^5+25 x^4+44 x^3+6 x^2+18 x+9$
• $y^2=12 x^6+8 x^5+x^4+43 x^3+25 x^2+5 x+10$
• $y^2=17 x^6+27 x^5+2 x^4+29 x^3+21 x^2+18 x+28$
• $y^2=4 x^6+20 x^5+26 x^4+16 x^3+42 x^2+12 x+25$
• $y^2=24 x^6+43 x^5+5 x^4+41 x^3+34 x^2+38 x+42$
• $y^2=18 x^6+18 x^5+20 x^4+9 x^3+9 x^2+21 x+23$
• $y^2=46 x^6+37 x^5+45 x^4+38 x^3+18 x^2+43 x+25$
• and

• $y^2=28 x^5+14 x^4+15 x^3+45 x^2+45 x+9$
• $y^2=18 x^6+19 x^5+6 x^4+14 x^3+2 x^2+5 x+19$
• $y^2=32 x^6+8 x^5+9 x^4+2 x^3+28 x^2+40 x+36$
• $y^2=36 x^6+39 x^5+10 x^4+37 x^3+7 x^2+30 x+27$
• $y^2=16 x^6+44 x^5+36 x^4+17 x^3+26 x+6$
• $y^2=14 x^6+14 x^5+28 x^4+2 x^3+41 x^2+5 x+9$
• $y^2=18 x^6+16 x^5+33 x^4+5 x^3+3 x^2+12 x+7$
• $y^2=12 x^6+36 x^5+7 x^4+29 x^3+19 x^2+44 x+13$
• $y^2=10 x^6+31 x^5+11 x^4+40 x^2+30 x+21$
• $y^2=34 x^6+28 x^5+8 x^4+30 x^3+15 x^2+30 x+20$
• $y^2=12 x^6+14 x^5+27 x^4+42 x^3+30 x^2+30 x+29$
• $y^2=44 x^6+18 x^5+4 x^4+26 x^3+36 x^2+20 x+12$
• $y^2=22 x^6+4 x^5+5 x^4+14 x^3+31 x^2+20 x+4$
• $y^2=6 x^6+36 x^5+34 x^4+14 x^3+5 x^2+26 x+42$
• $y^2=46 x^6+27 x^5+31 x^4+43 x^3+39 x^2+3 x+31$
• $y^2=12 x^6+43 x^5+28 x^4+17 x^3+44 x^2+17 x+30$
• $y^2=12 x^6+3 x^5+17 x^4+44 x^3+46 x^2+43 x+2$
• $y^2=9 x^6+20 x^5+46 x^4+22 x^3+6 x^2+4 x+19$
• $y^2=31 x^5+22 x^4+25 x^3+5 x^2+3 x+14$
• $y^2=4 x^6+16 x^5+32 x^4+46 x^3+9 x^2+46 x+9$
• $y^2=36 x^6+44 x^5+37 x^4+25 x^3+22 x^2+29 x+12$
• $y^2=25 x^6+14 x^5+27 x^4+35 x^3+32 x^2+23 x+35$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is 4.0.424400.1.

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.ao_fi	$2$	(not in LMFDB)