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

• Label: 2.59.a_p
• 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:	$1 + 15 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.270286448097$, $\pm0.729713551903$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{103}, \sqrt{-133})\)
Galois group:	$C_2^2$
Jacobians:	$36$
Isomorphism classes:	48
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3497$	$12229009$	$42180380372$	$146993776561801$	$511116754151460977$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3512$	$205380$	$12130836$	$714924300$	$42180227102$	$2488651484820$	$146830395299428$	$8662995818654940$	$511116755002280552$

Jacobians and polarizations

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

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

• $y^2=49 x^6+9 x^5+x^4+54 x^3+50 x^2+12 x+31$
• $y^2=34 x^6+12 x^5+56 x^4+12 x^3+29 x^2+29 x+4$
• $y^2=9 x^6+24 x^5+53 x^4+24 x^3+58 x^2+58 x+8$
• $y^2=3 x^6+31 x^5+4 x^4+4 x^3+24 x^2+18 x+30$
• $y^2=6 x^6+3 x^5+8 x^4+8 x^3+48 x^2+36 x+1$
• $y^2=29 x^6+3 x^5+55 x^4+13 x^3+25 x^2+42 x+30$
• $y^2=2 x^6+7 x^5+49 x^4+21 x^3+23 x^2+19 x+38$
• $y^2=34 x^6+12 x^5+44 x^4+51 x^3+28 x^2+20 x+4$
• $y^2=9 x^6+24 x^5+29 x^4+43 x^3+56 x^2+40 x+8$
• $y^2=39 x^6+57 x^5+40 x^4+44 x^3+47 x^2+19 x+18$
• $y^2=43 x^6+52 x^5+51 x^4+30 x^3+3 x^2+38 x+31$
• $y^2=27 x^6+45 x^5+43 x^4+x^3+6 x^2+17 x+3$
• $y^2=51 x^6+23 x^5+2 x^4+29 x^3+38 x^2+7 x+30$
• $y^2=43 x^6+46 x^5+4 x^4+58 x^3+17 x^2+14 x+1$
• $y^2=44 x^6+55 x^5+7 x^4+32 x^3+14 x^2+15 x+9$
• $y^2=29 x^6+51 x^5+14 x^4+5 x^3+28 x^2+30 x+18$

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 \(\Q(\sqrt{103}, \sqrt{-133})\).

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

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.p^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13699}) \)$)$

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_ap	$4$	(not in LMFDB)