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

• Label: 2.59.a_ady
• 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:	$1 - 102 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.0838471218745$, $\pm0.916152878126$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{55})\)
Galois group:	$C_2^2$
Jacobians:	$52$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3380$	$11424400$	$42180537620$	$146747057766400$	$511116754550304500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3278$	$205380$	$12110478$	$714924300$	$42180541598$	$2488651484820$	$146830462379038$	$8662995818654940$	$511116755799967598$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 52 curves (of which all are hyperelliptic):

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

• $y^2=25 x^6+29 x^5+41 x^4+38 x^3+8 x^2+5 x+39$
• $y^2=48 x^6+37 x^5+23 x^4+48 x^3+29 x^2+3 x+6$
• $y^2=37 x^6+15 x^5+46 x^4+37 x^3+58 x^2+6 x+12$
• $y^2=22 x^6+58 x^5+50 x^4+18 x^3+32 x^2+30 x+46$
• $y^2=44 x^6+57 x^5+41 x^4+36 x^3+5 x^2+x+33$
• $y^2=17 x^6+49 x^5+45 x^4+19 x^3+51 x^2+33 x$
• $y^2=34 x^6+39 x^5+31 x^4+38 x^3+43 x^2+7 x$
• $y^2=x^6+44 x^5+13 x^4+56 x^3+12 x^2+34 x+56$
• $y^2=28 x^6+11 x^5+38 x^4+50 x^3+15 x^2+46 x+26$
• $y^2=56 x^6+22 x^5+17 x^4+41 x^3+30 x^2+33 x+52$
• $y^2=25 x^6+20 x^5+9 x^4+22 x^2+32 x+53$
• $y^2=18 x^6+19 x^5+42 x^4+50 x^3+7 x^2+12 x+5$
• $y^2=25 x^6+28 x^4+5 x^3+44 x^2+43$
• $y^2=34 x^6+13 x^5+38 x^4+32 x^3+21 x+49$
• $y^2=9 x^6+26 x^5+17 x^4+5 x^3+42 x+39$
• $y^2=37 x^6+57 x^5+50 x^4+34 x^3+21 x^2+56 x+41$
• $y^2=15 x^6+55 x^5+41 x^4+9 x^3+42 x^2+53 x+23$
• $y^2=25 x^6+40 x^5+36 x^4+10 x^3+8 x^2+15$
• $y^2=16 x^6+31 x^5+35 x^4+7 x^3+10 x^2+49 x+23$
• $y^2=4 x^6+35 x^5+8 x^4+30 x^3+5 x^2+57 x+42$
• $y^2=34 x^6+24 x^5+13 x^4+29 x^2+14 x+45$
• $y^2=42 x^6+19 x^5+52 x^4+24 x^2+56 x+58$
• $y^2=6 x^6+18 x^5+58 x^4+14 x^3+49 x^2+30 x+41$
• $y^2=6 x^6+15 x^5+37 x^4+27 x^3+41 x^2+51 x+36$
• $y^2=12 x^6+17 x^5+15 x^4+49 x^2+58 x+3$
• $y^2=49 x^6+39 x^5+14 x^4+4 x^3+25 x^2+32 x+53$
• $y^2=2 x^6+36 x^5+54 x^4+43 x^3+42 x^2+32 x+51$
• $y^2=4 x^6+13 x^5+49 x^4+27 x^3+25 x^2+5 x+43$
• $y^2=31 x^6+53 x^5+39 x^4+33 x^2+42 x+52$
• $y^2=22 x^6+9 x^5+4 x^4+x^2+40 x+28$
• $y^2=29 x^6+6 x^5+11 x^4+50 x^3+35 x^2+43 x+51$
• $y^2=58 x^6+12 x^5+22 x^4+41 x^3+11 x^2+27 x+43$

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 \(\Q(i, \sqrt{55})\).

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

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.ady^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-55}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.59.ai_fe	$4$	(not in LMFDB)
2.59.a_dy	$4$	(not in LMFDB)
2.59.i_fe	$4$	(not in LMFDB)