Abelian variety isogeny class 2.113.az_ly over $\F_{113}$

• Label: 2.113.az_ly
• Base field: $\F_{113}$
• 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:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$( 1 - 21 x + 113 x^{2} )( 1 - 4 x + 113 x^{2} )$
	$1 - 25 x + 310 x^{2} - 2825 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0498602789898$, $\pm0.439752777404$
Angle rank:	$2$ (numerical)
Jacobians:	$54$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10230$	$162963900$	$2080725407640$	$26578043193240000$	$339449136723282780150$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$89$	$12765$	$1442048$	$163008113$	$18423939169$	$2081951266770$	$235260563787793$	$26584441808753953$	$3004041933992497424$	$339456738980401853325$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 54 curves (of which all are hyperelliptic):

• $y^2=78 x^6+6 x^5+110 x^4+12 x^3+91 x^2+21 x+46$
• $y^2=38 x^6+66 x^5+105 x^4+50 x^3+22 x^2+24 x+112$
• $y^2=60 x^6+25 x^5+55 x^4+76 x^3+83 x^2+89 x+46$
• $y^2=107 x^6+25 x^5+28 x^4+34 x^3+87 x^2+73 x+60$
• $y^2=24 x^6+110 x^5+75 x^4+76 x^3+5 x^2+88 x$
• $y^2=10 x^6+80 x^5+13 x^4+87 x^3+60 x^2+73 x+51$
• $y^2=29 x^6+83 x^5+93 x^4+106 x^3+101 x^2+12 x+1$
• $y^2=39 x^6+39 x^5+21 x^4+76 x^3+39 x^2+103 x+101$
• $y^2=64 x^6+11 x^5+22 x^4+5 x^3+78 x^2+78 x+92$
• $y^2=15 x^6+22 x^5+76 x^4+37 x^3+69 x^2+19 x+44$
• $y^2=50 x^6+100 x^5+91 x^4+78 x^3+98 x^2+13 x+100$
• $y^2=105 x^6+20 x^5+45 x^4+94 x^3+106 x^2+29 x+6$
• $y^2=109 x^6+39 x^5+14 x^4+35 x^3+85 x^2+8 x+87$
• $y^2=16 x^6+99 x^5+76 x^4+100 x^3+33 x^2+41 x+48$
• $y^2=21 x^6+61 x^5+84 x^4+66 x^3+48 x^2+90 x+37$
• $y^2=65 x^6+71 x^5+x^4+26 x^3+34 x^2+51 x+45$
• $y^2=79 x^6+39 x^5+29 x^4+15 x^3+54 x^2+86 x+54$
• $y^2=99 x^6+95 x^5+72 x^4+8 x^3+97 x+58$
• $y^2=100 x^6+26 x^5+73 x^4+24 x^3+103 x^2+91 x+46$
• $y^2=64 x^6+30 x^5+17 x^4+20 x^3+97 x^2+70 x+60$
• and

• $y^2=105 x^6+65 x^5+82 x^4+73 x^3+3 x^2+71 x+8$
• $y^2=65 x^6+88 x^5+91 x^4+2 x^3+71 x^2+61$
• $y^2=27 x^6+82 x^5+73 x^4+48 x^3+102 x^2+83 x+86$
• $y^2=48 x^6+61 x^5+60 x^4+59 x^3+31 x^2+58 x+34$
• $y^2=103 x^6+66 x^5+66 x^4+105 x^3+91 x^2+57 x+42$
• $y^2=47 x^6+11 x^5+53 x^4+100 x^3+28 x^2+19 x+12$
• $y^2=8 x^6+16 x^5+19 x^4+89 x^3+86 x^2+55 x+51$
• $y^2=13 x^6+68 x^5+49 x^4+71 x^3+89 x^2+3 x+20$
• $y^2=7 x^6+7 x^5+37 x^4+8 x^3+20 x^2+58 x+48$
• $y^2=83 x^6+68 x^5+66 x^4+11 x^3+60 x^2+111 x+56$
• $y^2=109 x^6+20 x^5+109 x^4+40 x^3+83 x^2+6 x+10$
• $y^2=10 x^6+59 x^5+43 x^4+20 x^3+50 x^2+x+24$
• $y^2=52 x^6+63 x^5+9 x^4+53 x^3+4 x^2+22 x+29$
• $y^2=19 x^6+5 x^5+19 x^4+111 x^3+59 x^2+97 x+75$
• $y^2=71 x^6+63 x^5+46 x^4+97 x^3+28 x^2+52 x+8$
• $y^2=100 x^6+33 x^5+96 x^4+12 x^3+72 x^2+89 x+37$
• $y^2=111 x^6+99 x^5+43 x^4+75 x^3+91 x^2+48 x+38$
• $y^2=27 x^6+x^5+50 x^4+50 x^3+76 x^2+98 x+70$
• $y^2=42 x^6+6 x^5+10 x^4+35 x^3+91 x^2+28 x+110$
• $y^2=89 x^6+37 x^5+14 x^4+13 x^3+50 x^2+107 x+10$
• $y^2=76 x^6+51 x^5+72 x^4+80 x^3+102 x^2+40 x+86$
• $y^2=91 x^6+109 x^5+88 x^4+70 x^3+21 x^2+88 x+6$
• $y^2=92 x^6+106 x^5+94 x^4+x^3+52 x^2+19 x+90$
• $y^2=55 x^6+58 x^5+5 x^4+87 x^3+58 x^2+43 x+103$
• $y^2=91 x^6+9 x^5+43 x^4+4 x^3+45 x^2+55 x+24$
• $y^2=8 x^6+8 x^5+23 x^4+82 x^3+8 x^2+104 x+21$
• $y^2=34 x^6+54 x^5+106 x^4+38 x^3+93 x^2+29 x+42$
• $y^2=101 x^6+65 x^5+81 x^4+18 x^3+57 x^2+49 x+86$
• $y^2=66 x^6+36 x^5+97 x^4+77 x^3+18 x^2+36 x+102$
• $y^2=21 x^6+81 x^5+48 x^4+80 x^3+79 x^2+43 x+110$
• $y^2=39 x^6+30 x^5+86 x^4+16 x^3+77 x^2+94 x+63$
• $y^2=x^6+76 x^5+92 x^4+73 x^3+76 x^2+29 x+82$
• $y^2=10 x^6+76 x^5+78 x^4+38 x^3+18 x^2+88 x+28$
• $y^2=71 x^6+57 x^5+53 x^4+52 x^3+15 x^2+84 x+31$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113}$.

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.av $\times$ 1.113.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.113.av : \(\Q(\sqrt{-11}) \).
• 1.113.ae : \(\Q(\sqrt{-109}) \).

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
2.113.ar_fm	$2$	(not in LMFDB)
2.113.r_fm	$2$	(not in LMFDB)
2.113.z_ly	$2$	(not in LMFDB)