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

• Label: 2.47.at_gq
• Base field: $\F_{47}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 - 13 x + 47 x^{2} )( 1 - 6 x + 47 x^{2} )$
	$1 - 19 x + 172 x^{2} - 893 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.102979434792$, $\pm0.355830380849$
Angle rank:	$2$ (numerical)
Jacobians:	$18$
Isomorphism classes:	150

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1470$	$4842180$	$10806810840$	$23810548557600$	$52594117115543850$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$29$	$2193$	$104090$	$4879529$	$229323139$	$10779101226$	$506623989733$	$23811303612721$	$1119130592070470$	$52599132608964393$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{47}$

The isogeny class factors as 1.47.an $\times$ 1.47.ag 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.an : \(\Q(\sqrt{-19}) \).
• 1.47.ag : \(\Q(\sqrt{-38}) \).

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.ah_q	$2$	(not in LMFDB)
2.47.h_q	$2$	(not in LMFDB)
2.47.t_gq	$2$	(not in LMFDB)