Abelian variety isogeny class 2.59.i_em over $\F_{59}$

• Label: 2.59.i_em
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 + 8 x + 116 x^{2} + 472 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.494972225260$, $\pm0.680272799183$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-202 +24 \sqrt{2}})\)
Galois group:	$D_{4}$
Jacobians:	$60$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4078$	$12715204$	$42005100526$	$146800234411408$	$511125396666630238$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$3650$	$204524$	$12114870$	$714936388$	$42180545234$	$2488654145836$	$146830417940638$	$8662995619736804$	$511116755876622530$

Jacobians and polarizations

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

• $y^2=44 x^6+7 x^5+39 x^4+23 x^3+53 x^2+3 x+30$
• $y^2=12 x^6+24 x^5+5 x^4+27 x^3+45 x^2+58 x+48$
• $y^2=24 x^6+29 x^5+47 x^4+34 x^3+14 x^2+58 x+3$
• $y^2=51 x^6+35 x^5+3 x^4+32 x^3+2 x^2+54 x+16$
• $y^2=32 x^6+43 x^5+11 x^3+x^2+27 x+15$
• $y^2=41 x^6+35 x^5+33 x^4+14 x^3+29 x^2+29 x+58$
• $y^2=22 x^6+19 x^5+17 x^4+53 x^3+43 x^2+48 x+54$
• $y^2=50 x^6+35 x^5+2 x^4+37 x^3+22 x^2+41 x+36$
• $y^2=4 x^6+27 x^5+45 x^4+6 x^3+56 x^2+55 x+24$
• $y^2=33 x^6+17 x^5+44 x^4+24 x^3+49 x^2+35 x+50$
• $y^2=13 x^6+31 x^5+17 x^4+26 x^3+34 x^2+37 x+28$
• $y^2=2 x^6+41 x^5+12 x^4+30 x^3+21 x^2+7 x+35$
• $y^2=27 x^6+6 x^5+31 x^4+56 x^3+50 x^2+30 x+50$
• $y^2=20 x^6+31 x^5+56 x^4+19 x^3+38 x^2+47$
• $y^2=31 x^6+13 x^5+52 x^4+36 x^3+48 x^2+x+19$
• $y^2=39 x^6+12 x^5+20 x^4+50 x^3+x^2+51 x+25$
• $y^2=5 x^6+41 x^5+20 x^4+37 x^3+18 x^2+24 x+11$
• $y^2=23 x^6+53 x^5+6 x^4+27 x^3+45 x^2+15 x+29$
• $y^2=3 x^6+44 x^5+35 x^4+16 x^3+58 x^2+13 x+25$
• $y^2=30 x^6+45 x^5+31 x^4+29 x^3+15 x^2+54 x+4$
• and

• $y^2=x^6+3 x^5+3 x^4+2 x^3+42 x^2+24 x+18$
• $y^2=30 x^6+21 x^5+11 x^4+34 x^3+56 x^2+6 x+56$
• $y^2=31 x^6+7 x^5+30 x^4+49 x^3+6 x^2+49 x+3$
• $y^2=7 x^6+47 x^5+38 x^4+15 x^3+43 x^2+30 x+43$
• $y^2=29 x^6+23 x^5+7 x^4+53 x^3+24 x^2+48 x+14$
• $y^2=53 x^6+6 x^5+13 x^4+55 x^3+29 x^2+57 x+11$
• $y^2=20 x^6+58 x^5+45 x^4+41 x^3+3 x^2+48 x+57$
• $y^2=33 x^6+6 x^5+43 x^4+22 x^3+25 x^2+37 x+48$
• $y^2=53 x^6+49 x^5+23 x^4+54 x^3+32 x^2+36 x+32$
• $y^2=48 x^6+32 x^5+6 x^4+53 x^3+17 x^2+42 x+52$
• $y^2=44 x^6+16 x^5+22 x^4+30 x^3+33 x^2+25 x+36$
• $y^2=40 x^6+6 x^5+13 x^4+32 x^3+27 x^2+5 x+13$
• $y^2=42 x^6+53 x^5+24 x^4+11 x^3+11 x^2+52 x+27$
• $y^2=57 x^6+32 x^5+17 x^4+41 x^3+33 x^2+15 x+2$
• $y^2=14 x^6+39 x^5+44 x^4+19 x^3+3 x^2+58 x+3$
• $y^2=40 x^6+45 x^5+54 x^4+19 x^3+52 x^2+10 x+16$
• $y^2=9 x^6+53 x^5+33 x^4+3 x^3+56 x^2+30 x+29$
• $y^2=15 x^6+23 x^5+42 x^4+42 x^3+57 x^2+30 x+29$
• $y^2=55 x^6+55 x^5+53 x^4+21 x^3+21 x^2+4 x+47$
• $y^2=22 x^6+34 x^5+17 x^4+8 x^3+21 x^2+44 x+34$
• $y^2=51 x^6+34 x^5+29 x^4+2 x^3+9 x^2+17 x+11$
• $y^2=31 x^6+58 x^5+29 x^4+55 x^3+37 x^2+25 x+16$
• $y^2=12 x^6+36 x^5+23 x^4+7 x^3+39 x^2+54 x+12$
• $y^2=10 x^6+47 x^5+26 x^4+6 x^3+58 x^2+57 x+57$
• $y^2=26 x^6+4 x^5+29 x^4+25 x^3+13 x^2+52 x+53$
• $y^2=14 x^6+29 x^5+26 x^4+34 x^3+5 x^2+47 x+36$
• $y^2=45 x^6+21 x^5+28 x^4+24 x^3+8 x^2+12 x+9$
• $y^2=32 x^6+2 x^5+31 x^4+32 x^3+55 x^2+6 x+57$
• $y^2=34 x^5+49 x^4+35 x^3+12 x^2+8 x+29$
• $y^2=24 x^6+21 x^5+15 x^4+49 x^3+12 x^2+24 x+19$
• $y^2=2 x^6+57 x^5+x^4+23 x^3+4 x^2+55 x+16$
• $y^2=46 x^6+43 x^5+15 x^4+13 x^3+54 x^2+x+22$
• $y^2=9 x^6+52 x^5+7 x^4+18 x^3+52 x^2+54 x+29$
• $y^2=27 x^6+48 x^5+40 x^4+40 x^3+27 x^2+18 x+49$
• $y^2=28 x^6+37 x^5+19 x^4+28 x^3+37 x+5$
• $y^2=55 x^6+16 x^5+42 x^4+47 x^3+41 x^2+24 x+33$
• $y^2=30 x^6+18 x^5+35 x^4+32 x^3+31 x^2+51 x+12$
• $y^2=12 x^6+27 x^5+7 x^4+28 x^3+55 x^2+44 x+34$
• $y^2=24 x^6+29 x^5+51 x^3+57 x^2+37 x+32$
• $y^2=17 x^6+25 x^5+45 x^4+37 x^3+27 x^2+45 x+23$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-202 +24 \sqrt{2}})\).

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
2.59.ai_em	$2$	(not in LMFDB)