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

• Label: 2.47.aj_bi
• 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:	$1 - 9 x + 34 x^{2} - 423 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.0612517683063$, $\pm0.605414898360$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-107})\)
Galois group:	$C_2^2$
Jacobians:	$57$
Isomorphism classes:	126
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1812$	$4848912$	$10667584656$	$23790566067264$	$52602081778038252$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$2197$	$102744$	$4875433$	$229357869$	$10779047422$	$506621760411$	$23811294956401$	$1119130494561288$	$52599131942545357$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 57 curves (of which all are hyperelliptic):

• $y^2=19 x^6+9 x^5+36 x^4+31 x^3+6 x^2+29 x+13$
• $y^2=40 x^6+30 x^5+32 x^4+23 x^2+40 x+3$
• $y^2=18 x^6+24 x^5+10 x^4+11 x^3+8 x^2+23 x+43$
• $y^2=14 x^6+26 x^5+x^4+2 x^3+26 x^2+3 x+36$
• $y^2=11 x^6+8 x^5+34 x^4+2 x^3+43 x^2+2 x+35$
• $y^2=5 x^6+28 x^5+19 x^3+38 x^2+34 x+3$
• $y^2=24 x^6+42 x^5+4 x^4+7 x^3+28 x^2+13$
• $y^2=29 x^6+14 x^5+43 x^4+27 x^3+29 x^2+23 x+34$
• $y^2=x^6+34 x^5+31 x^4+12 x^3+9 x^2+3 x+27$
• $y^2=x^6+41 x^5+10 x^4+24 x^3+15 x^2+31 x+41$
• $y^2=4 x^6+20 x^5+25 x^4+2 x^3+11 x^2+25 x+30$
• $y^2=13 x^6+17 x^5+43 x^4+x^3+17 x^2+43 x+5$
• $y^2=39 x^6+32 x^5+21 x^4+10 x^3+29 x^2+17 x+15$
• $y^2=14 x^6+30 x^5+9 x^4+8 x^3+22 x^2+7 x+14$
• $y^2=20 x^6+20 x^5+33 x^4+27 x^2+2 x+33$
• $y^2=32 x^6+18 x^5+36 x^4+17 x^3+35 x^2+18 x+5$
• $y^2=34 x^6+25 x^5+16 x^3+14 x^2+9 x+12$
• $y^2=8 x^6+7 x^5+9 x^3+27 x^2+24 x+38$
• $y^2=12 x^6+27 x^5+33 x^4+46 x^3+26 x^2+10 x+43$
• $y^2=14 x^6+10 x^5+3 x^4+x^3+13 x^2+35 x+23$
• and

• $y^2=27 x^6+32 x^5+37 x^4+41 x^3+13 x^2+14 x+39$
• $y^2=34 x^6+43 x^5+23 x^4+25 x^3+28 x^2+13 x+39$
• $y^2=28 x^6+32 x^5+24 x^4+36 x^3+33 x^2+44$
• $y^2=x^6+4 x^5+28 x^4+27 x^3+38 x^2+39 x+24$
• $y^2=22 x^6+36 x^5+45 x^4+35 x^3+31 x^2+3 x+14$
• $y^2=35 x^6+33 x^5+43 x^4+19 x^3+16 x^2+12 x+43$
• $y^2=13 x^6+23 x^5+33 x^4+7 x^3+36 x^2+37 x+45$
• $y^2=21 x^6+39 x^5+20 x^4+38 x^3+43 x^2+24 x+20$
• $y^2=25 x^6+21 x^5+41 x^4+33 x^3+38 x^2+10 x+18$
• $y^2=43 x^6+9 x^5+45 x^4+6 x^3+32 x^2+39 x+13$
• $y^2=21 x^6+28 x^5+6 x^4+25 x^3+14 x^2+18 x+22$
• $y^2=10 x^6+19 x^5+6 x^4+17 x^3+39 x^2+29 x+41$
• $y^2=24 x^6+41 x^5+25 x^4+32 x^3+42 x^2+25 x+12$
• $y^2=35 x^6+44 x^5+38 x^4+40 x^3+30 x+27$
• $y^2=46 x^6+25 x^5+11 x^4+19 x^3+10 x^2+36 x+23$
• $y^2=39 x^6+35 x^5+8 x^3+29 x^2+24 x+20$
• $y^2=2 x^6+32 x^5+6 x^4+13 x^3+17 x^2+27 x+2$
• $y^2=7 x^6+23 x^5+4 x^4+12 x^3+19 x^2+16 x+33$
• $y^2=3 x^6+32 x^5+14 x^4+36 x^3+36 x^2+5 x+31$
• $y^2=43 x^6+12 x^5+22 x^4+29 x^3+20 x^2+4 x+21$
• $y^2=40 x^6+6 x^4+4 x^3+11 x^2+40 x+34$
• $y^2=17 x^6+7 x^5+3 x^4+3 x^3+13 x^2+16 x+19$
• $y^2=16 x^6+6 x^5+26 x^4+39 x^3+x^2+2 x+43$
• $y^2=19 x^5+13 x^4+14 x^3+2 x^2+11 x+14$
• $y^2=45 x^6+30 x^5+22 x^4+9 x^3+17 x^2+x+1$
• $y^2=45 x^6+21 x^5+43 x^4+21 x^3+2 x^2+25 x+36$
• $y^2=41 x^6+3 x^5+17 x^4+35 x^3+29 x^2+32 x+10$
• $y^2=41 x^6+32 x^5+11 x^4+14 x^3+x^2+10 x+17$
• $y^2=43 x^6+9 x^5+3 x^4+5 x^3+34 x^2+17 x+13$
• $y^2=4 x^6+46 x^5+12 x^4+40 x^3+33 x^2+26 x+26$
• $y^2=12 x^6+24 x^5+24 x^4+x^3+37 x^2+x+12$
• $y^2=43 x^6+9 x^5+29 x^4+46 x^3+4 x^2+9 x+11$
• $y^2=35 x^6+30 x^5+8 x^4+40 x^3+3 x^2+43 x+28$
• $y^2=18 x^6+x^5+x^4+16 x^3+x^2+30 x+15$
• $y^2=46 x^6+26 x^5+46 x^4+46 x^3+44 x^2+22 x+23$
• $y^2=25 x^6+37 x^5+6 x^4+3 x^3+43 x^2+23 x$
• $y^2=38 x^6+11 x^5+43 x^4+12 x^3+14 x^2+4 x+20$

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 \(\Q(\sqrt{-3}, \sqrt{-107})\).

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

The base change of $A$ to $\F_{47^{3}}$ is 1.103823.auu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-107}) \)$)$

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.j_bi	$2$	(not in LMFDB)
2.47.s_gt	$3$	(not in LMFDB)
2.47.as_gt	$6$	(not in LMFDB)