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

• Label: 2.59.v_im
• 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:	$1 + 21 x + 220 x^{2} + 1239 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.665389865024$, $\pm0.836179434257$
Angle rank:	$2$ (numerical)
Number field:	4.0.11072952.1
Galois group:	$D_{4}$
Jacobians:	$36$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4962$	$12117204$	$41999777208$	$146911016986272$	$511104307366637022$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$3481$	$204498$	$12124009$	$714906891$	$42180533170$	$2488650159321$	$146830463975185$	$8662995617662134$	$511116753363836761$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=22 x^6+45 x^5+50 x^4+4 x^3+22 x^2+52 x+28$
• $y^2=45 x^6+12 x^5+27 x^4+29 x^3+25 x^2+35 x+39$
• $y^2=45 x^6+43 x^5+25 x^4+50 x^3+39 x^2+33 x+57$
• $y^2=49 x^6+42 x^5+20 x^4+36 x^3+16 x^2+34 x+7$
• $y^2=4 x^6+38 x^5+52 x^4+30 x^3+25 x^2+7 x+35$
• $y^2=9 x^6+44 x^5+36 x^4+46 x^3+52 x^2+18 x+40$
• $y^2=4 x^6+49 x^5+36 x^4+49 x^3+16 x^2+26 x+10$
• $y^2=44 x^6+43 x^5+48 x^4+5 x^3+17 x^2+22 x+13$
• $y^2=26 x^6+14 x^5+43 x^4+53 x^3+27 x^2+16 x+7$
• $y^2=27 x^6+12 x^5+30 x^4+4 x^3+23 x^2+14 x+28$
• $y^2=20 x^6+11 x^5+31 x^4+27 x^3+9 x^2+23 x+2$
• $y^2=26 x^6+18 x^5+35 x^4+9 x^3+26 x^2+16 x+33$
• $y^2=7 x^6+49 x^5+20 x^3+32 x^2+32 x+54$
• $y^2=51 x^6+39 x^5+41 x^4+35 x^3+58 x^2+7 x+36$
• $y^2=39 x^6+50 x^5+14 x^4+31 x^3+31 x^2+9 x+30$
• $y^2=18 x^6+19 x^5+28 x^4+x^3+15 x^2+2 x+41$
• $y^2=10 x^6+22 x^5+26 x^4+42 x^3+50 x^2+9 x+25$
• $y^2=50 x^6+9 x^5+41 x^4+37 x^3+45 x^2+3 x+19$
• $y^2=32 x^6+43 x^5+24 x^4+6 x^3+36 x^2+15 x+58$
• $y^2=52 x^6+33 x^5+24 x^4+26 x^3+27 x^2+4 x+18$
• and

• $y^2=33 x^6+x^5+52 x^4+52 x^3+35 x^2+11 x+25$
• $y^2=27 x^6+47 x^5+11 x^4+15 x^3+33 x^2+28 x+49$
• $y^2=41 x^6+11 x^5+34 x^4+42 x^3+55 x+33$
• $y^2=17 x^6+48 x^5+13 x^4+37 x^3+32 x^2+58 x+20$
• $y^2=12 x^6+46 x^5+31 x^4+12 x^3+9 x^2+17 x+15$
• $y^2=53 x^5+20 x^4+11 x^3+12 x^2+38 x+3$
• $y^2=25 x^6+56 x^5+42 x^4+35 x^3+31 x^2+34 x+28$
• $y^2=25 x^6+25 x^5+16 x^4+43 x^3+29 x^2+27 x+19$
• $y^2=20 x^6+38 x^4+54 x^3+10 x^2+25 x+56$
• $y^2=9 x^6+40 x^5+45 x^4+34 x^3+25 x^2+2 x+51$
• $y^2=7 x^6+5 x^5+44 x^4+42 x^3+9 x^2+21 x+53$
• $y^2=14 x^6+12 x^5+43 x^4+41 x^3+19 x^2+23 x+46$
• $y^2=29 x^6+21 x^5+7 x^4+44 x^3+32 x^2+6 x+1$
• $y^2=41 x^6+34 x^5+49 x^4+56 x^3+23 x^2+23 x+27$
• $y^2=25 x^6+57 x^5+13 x^4+55 x^3+46 x^2+26 x+18$
• $y^2=28 x^6+31 x^5+5 x^4+12 x^3+9 x^2+18 x+12$

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 4.0.11072952.1.

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
2.59.av_im	$2$	(not in LMFDB)