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

• Label: 2.47.al_dq
• Base field: $\F_{47}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$( 1 - 11 x + 47 x^{2} )( 1 + 47 x^{2} )$
	$1 - 11 x + 94 x^{2} - 517 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.203632126579$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$60$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1776$	$5029632$	$10802264256$	$23807722834944$	$52606067858216976$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$2277$	$104044$	$4878953$	$229375247$	$10779582222$	$506623450937$	$23811273053041$	$1119130415227588$	$52599132238812957$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{2}}$.

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.al $\times$ 1.47.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.47.al : \(\Q(\sqrt{-67}) \).
• 1.47.a : \(\Q(\sqrt{-47}) \).

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

The base change of $A$ to $\F_{47^{2}}$ is 1.2209.abb $\times$ 1.2209.dq. The endomorphism algebra for each factor is:

• 1.2209.abb : \(\Q(\sqrt{-67}) \).
• 1.2209.dq : the quaternion algebra over \(\Q\) ramified at $47$ and $\infty$.

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.l_dq	$2$	(not in LMFDB)