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

• Label: 2.59.a_em
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 + 116 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.470655653043$, $\pm0.529344346957$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-13})\)
Galois group:	$C_2^2$
Jacobians:	$49$
Cyclic group of points:	yes

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3598$	$12945604$	$42180883150$	$146673123713424$	$511116754164732478$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3714$	$205380$	$12104374$	$714924300$	$42181232658$	$2488651484820$	$146830401729694$	$8662995818654940$	$511116755028823554$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 49 curves (of which all are hyperelliptic):

• $y^2=22 x^6+35 x^5+26 x^4+46 x^3+50 x^2+4 x+19$
• $y^2=44 x^6+11 x^5+52 x^4+33 x^3+41 x^2+8 x+38$
• $y^2=18 x^6+58 x^5+13 x^4+26 x^2+4 x+26$
• $y^2=x^6+2 x^5+6 x^4+12 x^2+51 x+8$
• $y^2=51 x^6+17 x^5+33 x^4+24 x^3+7 x^2+27 x+32$
• $y^2=43 x^6+34 x^5+7 x^4+48 x^3+14 x^2+54 x+5$
• $y^2=29 x^6+21 x^5+41 x^4+20 x^2+13 x+26$
• $y^2=58 x^6+42 x^5+23 x^4+40 x^2+26 x+52$
• $y^2=48 x^6+32 x^5+8 x^4+16 x^2+49 x+30$
• $y^2=10 x^6+11 x^5+54 x^4+34 x^3+54 x^2+40 x+13$
• $y^2=20 x^6+22 x^5+49 x^4+9 x^3+49 x^2+21 x+26$
• $y^2=6 x^6+49 x^5+15 x^4+30 x^2+40 x+48$
• $y^2=27 x^6+27 x^5+34 x^4+14 x^3+51 x^2+46 x+36$
• $y^2=54 x^6+54 x^5+9 x^4+28 x^3+43 x^2+33 x+13$
• $y^2=12 x^6+12 x^5+25 x^4+21 x^3+52 x^2+51 x+44$
• $y^2=24 x^6+24 x^5+50 x^4+42 x^3+45 x^2+43 x+29$
• $y^2=52 x^6+10 x^5+7 x^4+35 x^3+14 x^2+7 x+9$
• $y^2=45 x^6+20 x^5+14 x^4+11 x^3+28 x^2+14 x+18$
• $y^2=18 x^6+25 x^5+39 x^4+50 x^3+38 x+34$
• $y^2=36 x^6+50 x^5+19 x^4+41 x^3+17 x+9$
• and

• $y^2=13 x^6+18 x^5+11 x^4+28 x^3+28 x^2+48 x+37$
• $y^2=26 x^6+36 x^5+22 x^4+56 x^3+56 x^2+37 x+15$
• $y^2=40 x^6+8 x^5+44 x^4+21 x^3+53 x^2+17 x+4$
• $y^2=21 x^6+16 x^5+29 x^4+42 x^3+47 x^2+34 x+8$
• $y^2=25 x^6+4 x^5+12 x^4+46 x^3+7 x^2+13 x+50$
• $y^2=50 x^6+8 x^5+24 x^4+33 x^3+14 x^2+26 x+41$
• $y^2=27 x^6+19 x^5+33 x^4+53 x^3+11 x^2+35 x+4$
• $y^2=54 x^6+38 x^5+7 x^4+47 x^3+22 x^2+11 x+8$
• $y^2=27 x^6+27 x^5+23 x^4+19 x^3+18 x^2+46 x+26$
• $y^2=54 x^6+54 x^5+46 x^4+38 x^3+36 x^2+33 x+52$
• $y^2=45 x^6+32 x^5+4 x^4+40 x^3+53 x^2+48 x+32$
• $y^2=31 x^6+5 x^5+8 x^4+21 x^3+47 x^2+37 x+5$
• $y^2=9 x^6+4 x^5+34 x^4+24 x^3+21 x^2+27 x+24$
• $y^2=18 x^6+8 x^5+9 x^4+48 x^3+42 x^2+54 x+48$
• $y^2=22 x^6+18 x^5+16 x^4+56 x^3+4 x^2+15 x+16$
• $y^2=44 x^6+36 x^5+32 x^4+53 x^3+8 x^2+30 x+32$
• $y^2=6 x^6+47 x^5+41 x^4+15 x^3+46 x^2+10 x+11$
• $y^2=12 x^6+35 x^5+23 x^4+30 x^3+33 x^2+20 x+22$
• $y^2=51 x^6+31 x^5+2 x^4+40 x^3+x^2+21 x+17$
• $y^2=43 x^6+3 x^5+4 x^4+21 x^3+2 x^2+42 x+34$
• $y^2=2 x^5+55 x^4+54 x^3+44 x^2+2 x+2$
• $y^2=4 x^5+51 x^4+49 x^3+29 x^2+4 x+4$
• $y^2=29 x^6+36 x^5+50 x^4+17 x^3+5 x^2+51 x+24$
• $y^2=58 x^6+13 x^5+41 x^4+34 x^3+10 x^2+43 x+48$
• $y^2=13 x^6+58 x^5+4 x^4+8 x^2+4 x+45$
• $y^2=49 x^6+49 x^5+28 x^4+46 x^3+39 x^2+11 x+28$
• $y^2=39 x^6+39 x^5+56 x^4+33 x^3+19 x^2+22 x+56$
• $y^2=27 x^6+6 x^5+27 x^4+12 x^3+4 x^2+19 x+10$
• $y^2=54 x^6+12 x^5+54 x^4+24 x^3+8 x^2+38 x+20$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{2}}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-13})\).

Endomorphism algebra over $\overline{\F}_{59}$

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.em^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

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.a_aem	$4$	(not in LMFDB)
2.59.ac_c	$8$	(not in LMFDB)
2.59.c_c	$8$	(not in LMFDB)