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

• Label: 2.59.i_fc
• 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:	$1 + 8 x + 132 x^{2} + 472 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.553834308828$, $\pm0.614646336025$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-14 +4 \sqrt{2}})\)
Galois group:	$D_{4}$
Jacobians:	$34$
Isomorphism classes:	34
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4094$	$12830596$	$41926625342$	$146753739606416$	$511182592763200174$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$3682$	$204140$	$12111030$	$715016388$	$42180521842$	$2488646728748$	$146830455786654$	$8662995986009060$	$511116751847682402$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 34 curves (of which all are hyperelliptic):

• $y^2=52 x^6+18 x^5+18 x^4+55 x^3+44 x^2+3 x+18$
• $y^2=15 x^6+41 x^5+8 x^4+2 x^3+17 x^2+54 x+26$
• $y^2=27 x^6+41 x^5+57 x^4+10 x^3+43 x^2+42$
• $y^2=48 x^6+46 x^5+20 x^4+48 x^3+28 x^2+22$
• $y^2=33 x^6+12 x^5+37 x^4+35 x^3+56 x^2+6 x+5$
• $y^2=49 x^6+28 x^5+12 x^4+18 x^3+42 x^2+51 x+51$
• $y^2=43 x^6+52 x^5+51 x^4+12 x^3+57 x^2+22 x+16$
• $y^2=49 x^6+35 x^5+52 x^4+24 x^3+21 x^2+46 x$
• $y^2=46 x^6+20 x^5+13 x^4+47 x^3+25 x^2+17 x+49$
• $y^2=25 x^6+x^5+15 x^4+7 x^3+13 x^2+20 x+26$
• $y^2=8 x^6+54 x^5+23 x^4+52 x^3+26 x^2+16 x+15$
• $y^2=4 x^6+11 x^5+8 x^4+7 x^3+34 x+26$
• $y^2=23 x^6+56 x^5+44 x^4+33 x^3+43 x^2+23 x+9$
• $y^2=44 x^6+56 x^5+8 x^4+48 x^3+41 x^2+3 x+37$
• $y^2=12 x^6+23 x^5+38 x^4+11 x^3+35 x^2+45 x+4$
• $y^2=21 x^6+46 x^5+45 x^4+27 x^3+8 x^2+46 x+19$
• $y^2=6 x^6+39 x^5+41 x^4+51 x^3+33 x^2+16 x+8$
• $y^2=14 x^6+3 x^5+40 x^4+52 x^3+13 x^2+8 x+28$
• $y^2=16 x^6+36 x^5+51 x^4+44 x^3+55 x^2+19 x+49$
• $y^2=17 x^6+25 x^5+25 x^4+14 x^3+50 x^2+36 x+41$
• and

• $y^2=27 x^6+27 x^5+30 x^4+24 x^3+31 x^2+50 x$
• $y^2=8 x^6+55 x^5+23 x^4+19 x^3+34 x^2+31 x+6$
• $y^2=38 x^6+58 x^5+39 x^4+50 x^3+11 x^2+50 x+46$
• $y^2=8 x^6+20 x^5+44 x^4+57 x^3+16 x^2+35 x+31$
• $y^2=20 x^6+45 x^5+45 x^4+11 x^3+33 x^2+38 x+5$
• $y^2=55 x^5+22 x^4+53 x^3+53 x^2+5 x+57$
• $y^2=40 x^6+42 x^5+32 x^4+57 x^3+28 x^2+6 x+24$
• $y^2=35 x^6+53 x^5+2 x^4+21 x^3+52 x^2+46 x+53$
• $y^2=25 x^6+56 x^5+36 x^4+49 x^3+11 x^2+36 x+57$
• $y^2=6 x^6+43 x^5+32 x^4+12 x^3+19 x^2+33 x+19$
• $y^2=36 x^6+54 x^5+46 x^4+7 x^3+x^2+46 x+37$
• $y^2=31 x^6+18 x^5+56 x^4+35 x^3+41 x^2+5 x+53$
• $y^2=27 x^6+x^5+43 x^4+19 x^3+22 x^2+49 x+6$
• $y^2=19 x^6+7 x^5+21 x^4+33 x^3+16 x^2+10 x+46$

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 \(\Q(\sqrt{-14 +4 \sqrt{2}})\).

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_fc	$2$	(not in LMFDB)