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

• Label: 2.113.az_mk
• Base field: $\F_{113}$
• 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:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$1 - 25 x + 322 x^{2} - 2825 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.0979274132770$, $\pm0.428463115496$
Angle rank:	$2$ (numerical)
Number field:	4.0.1034074124.1
Galois group:	$D_{4}$
Jacobians:	$50$
Isomorphism classes:	50

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10242$	$163277964$	$2082024608904$	$26580461263174656$	$339452066155010346882$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$89$	$12789$	$1442948$	$163022945$	$18424098169$	$2081953172898$	$235260595602793$	$26584442255971009$	$3004041938094118724$	$339456739005920244789$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 50 curves (of which all are hyperelliptic):

• $y^2=9 x^6+28 x^5+111 x^4+61 x^3+29 x^2+75 x+10$
• $y^2=14 x^6+59 x^5+70 x^4+12 x^3+66 x^2+101 x+37$
• $y^2=61 x^6+3 x^5+41 x^4+33 x^3+78 x^2+56 x+27$
• $y^2=71 x^6+34 x^5+82 x^4+50 x^3+18 x^2+x+86$
• $y^2=34 x^6+21 x^5+63 x^4+9 x^3+8 x^2+49 x+23$
• $y^2=87 x^5+5 x^4+59 x^3+26 x^2+48 x+74$
• $y^2=6 x^6+32 x^5+81 x^4+45 x^3+47 x^2+63 x+100$
• $y^2=37 x^6+107 x^5+14 x^4+110 x^3+84 x^2+78 x+43$
• $y^2=34 x^6+78 x^5+24 x^4+5 x^3+49 x^2+65 x+80$
• $y^2=69 x^6+15 x^5+10 x^4+69 x^3+86 x^2+76 x+75$
• $y^2=39 x^6+35 x^5+98 x^4+84 x^3+90 x^2+50 x+102$
• $y^2=45 x^6+77 x^5+74 x^4+50 x^3+101 x^2+3 x+89$
• $y^2=3 x^6+12 x^5+75 x^4+83 x^3+93 x^2+107 x+43$
• $y^2=20 x^6+89 x^5+101 x^4+80 x^3+41 x^2+82 x+92$
• $y^2=101 x^6+110 x^5+19 x^4+99 x^3+96 x^2+19 x+29$
• $y^2=81 x^6+25 x^5+68 x^4+13 x^3+x^2+15 x+37$
• $y^2=85 x^6+48 x^5+69 x^4+49 x^3+26 x^2+74 x+19$
• $y^2=62 x^6+56 x^5+42 x^3+66 x^2+43 x+41$
• $y^2=13 x^6+21 x^5+57 x^4+54 x^3+91 x^2+6 x+27$
• $y^2=71 x^6+66 x^5+26 x^4+90 x^3+74 x^2+73 x+60$
• and

• $y^2=38 x^6+107 x^5+82 x^4+3 x^3+57 x^2+34 x+86$
• $y^2=54 x^6+95 x^5+48 x^4+46 x^3+79 x^2+32 x+43$
• $y^2=85 x^6+62 x^5+27 x^4+99 x^3+102 x^2+107 x+43$
• $y^2=86 x^6+35 x^5+40 x^4+47 x^3+95 x^2+29$
• $y^2=93 x^6+110 x^5+22 x^4+103 x^3+69 x^2+57 x+46$
• $y^2=82 x^6+105 x^5+52 x^4+83 x^3+10 x^2+92 x+36$
• $y^2=11 x^6+100 x^5+18 x^4+57 x^3+18 x^2+102 x+63$
• $y^2=104 x^6+96 x^5+4 x^4+9 x^3+16 x^2+38 x+1$
• $y^2=61 x^6+2 x^5+x^4+86 x^3+101 x^2+39 x+11$
• $y^2=78 x^6+73 x^5+73 x^4+39 x^3+67 x^2+62 x+87$
• $y^2=64 x^6+27 x^5+49 x^4+51 x^3+76 x^2+73 x+39$
• $y^2=57 x^6+54 x^5+111 x^4+42 x^3+6 x^2+64 x+52$
• $y^2=57 x^6+99 x^5+62 x^4+94 x^3+31 x^2+8 x+12$
• $y^2=93 x^6+111 x^5+109 x^4+58 x^3+77 x^2+101 x+37$
• $y^2=16 x^6+57 x^5+84 x^4+58 x^3+89 x^2+22 x+107$
• $y^2=60 x^6+111 x^5+58 x^4+109 x^3+22 x^2+56 x+24$
• $y^2=24 x^6+34 x^5+4 x^4+88 x^3+4 x^2+5 x+47$
• $y^2=101 x^6+18 x^5+55 x^4+9 x^3+107 x^2+65 x+80$
• $y^2=75 x^6+45 x^5+19 x^4+24 x^3+100 x^2+61 x+59$
• $y^2=55 x^6+109 x^5+26 x^4+28 x^3+91 x^2+41$
• $y^2=51 x^6+105 x^5+61 x^4+99 x^3+97 x^2+81 x+5$
• $y^2=62 x^6+93 x^5+59 x^4+43 x^3+61 x^2+92 x+60$
• $y^2=89 x^6+43 x^5+29 x^4+19 x^3+89 x^2+109 x+3$
• $y^2=103 x^6+10 x^5+12 x^4+78 x^3+62 x^2+35 x+82$
• $y^2=75 x^6+60 x^5+34 x^4+75 x^3+19 x^2+81 x+55$
• $y^2=16 x^6+83 x^5+91 x^4+82 x^3+50 x^2+94 x+39$
• $y^2=33 x^6+48 x^5+25 x^4+69 x^3+32 x^2+27 x+7$
• $y^2=46 x^6+41 x^5+19 x^3+4 x^2+5 x+17$
• $y^2=105 x^6+75 x^5+28 x^4+59 x^3+12 x^2+77 x+60$
• $y^2=40 x^6+26 x^5+32 x^4+96 x^3+82 x^2+34 x+38$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{113}$

The endomorphism algebra of this simple isogeny class is 4.0.1034074124.1.

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
2.113.z_mk	$2$	(not in LMFDB)