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

• Label: 2.59.a_do
• 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:	$1 + 92 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.392304302513$, $\pm0.607695697487$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{26}, \sqrt{-210})\)
Galois group:	$C_2^2$
Jacobians:	$56$
Isomorphism classes:	128

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3574$	$12773476$	$42180351574$	$146794063539600$	$511116751912378054$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3666$	$205380$	$12114358$	$714924300$	$42180169506$	$2488651484820$	$146830481561758$	$8662995818654940$	$511116750524114706$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 56 curves (of which all are hyperelliptic):

• $y^2=3 x^6+32 x^5+28 x^4+41 x^3+30 x^2+28 x+50$
• $y^2=6 x^6+5 x^5+56 x^4+23 x^3+x^2+56 x+41$
• $y^2=18 x^6+21 x^5+4 x^4+32 x^3+50 x^2+48 x+56$
• $y^2=36 x^6+42 x^5+8 x^4+5 x^3+41 x^2+37 x+53$
• $y^2=35 x^6+21 x^5+26 x^4+52 x^3+51 x^2+19 x+50$
• $y^2=11 x^6+42 x^5+52 x^4+45 x^3+43 x^2+38 x+41$
• $y^2=37 x^6+14 x^5+11 x^4+27 x^3+2 x^2+23 x+43$
• $y^2=15 x^6+28 x^5+22 x^4+54 x^3+4 x^2+46 x+27$
• $y^2=58 x^6+11 x^5+31 x^4+8 x^3+35 x^2+3 x+32$
• $y^2=57 x^6+22 x^5+3 x^4+16 x^3+11 x^2+6 x+5$
• $y^2=3 x^6+12 x^5+45 x^4+22 x^3+20 x^2+33 x+30$
• $y^2=6 x^6+24 x^5+31 x^4+44 x^3+40 x^2+7 x+1$
• $y^2=28 x^6+23 x^4+27 x^3+36 x^2+24 x+23$
• $y^2=56 x^6+46 x^4+54 x^3+13 x^2+48 x+46$
• $y^2=39 x^6+19 x^5+56 x^4+35 x^3+48 x^2+51 x+44$
• $y^2=19 x^6+38 x^5+53 x^4+11 x^3+37 x^2+43 x+29$
• $y^2=56 x^6+29 x^5+15 x^3+18 x^2+32 x+5$
• $y^2=53 x^6+58 x^5+30 x^3+36 x^2+5 x+10$
• $y^2=22 x^6+52 x^5+14 x^4+34 x^3+33 x^2+40 x+47$
• $y^2=44 x^6+45 x^5+28 x^4+9 x^3+7 x^2+21 x+35$
• and

• $y^2=33 x^6+50 x^5+53 x^4+29 x^3+51 x^2+6 x+31$
• $y^2=7 x^6+41 x^5+47 x^4+58 x^3+43 x^2+12 x+3$
• $y^2=36 x^6+38 x^5+18 x^4+26 x^3+52 x^2+51 x+17$
• $y^2=13 x^6+17 x^5+36 x^4+52 x^3+45 x^2+43 x+34$
• $y^2=31 x^6+12 x^5+33 x^4+9 x^3+27 x^2+5 x+24$
• $y^2=3 x^6+24 x^5+7 x^4+18 x^3+54 x^2+10 x+48$
• $y^2=31 x^6+46 x^5+31 x^4+17 x^3+49 x^2+13 x+53$
• $y^2=3 x^6+33 x^5+3 x^4+34 x^3+39 x^2+26 x+47$
• $y^2=21 x^6+16 x^5+34 x^4+17 x^3+30 x^2+18 x+36$
• $y^2=42 x^6+32 x^5+9 x^4+34 x^3+x^2+36 x+13$
• $y^2=27 x^6+34 x^5+52 x^4+5 x^3+15 x^2+24 x+23$
• $y^2=54 x^6+9 x^5+45 x^4+10 x^3+30 x^2+48 x+46$
• $y^2=46 x^6+11 x^5+23 x^4+39 x^3+16 x^2+x+15$
• $y^2=33 x^6+22 x^5+46 x^4+19 x^3+32 x^2+2 x+30$
• $y^2=34 x^5+12 x^4+17 x^3+5 x^2+28 x+56$
• $y^2=9 x^5+24 x^4+34 x^3+10 x^2+56 x+53$
• $y^2=41 x^6+25 x^5+5 x^4+56 x^3+19 x^2+36 x+25$
• $y^2=23 x^6+50 x^5+10 x^4+53 x^3+38 x^2+13 x+50$
• $y^2=40 x^6+x^5+23 x^4+x^3+38 x^2+45 x+33$
• $y^2=21 x^6+2 x^5+46 x^4+2 x^3+17 x^2+31 x+7$
• $y^2=21 x^6+4 x^5+33 x^4+52 x^3+x^2+26 x+1$
• $y^2=42 x^6+8 x^5+7 x^4+45 x^3+2 x^2+52 x+2$
• $y^2=38 x^6+49 x^5+7 x^4+26 x^3+34 x^2+36 x+50$
• $y^2=17 x^6+39 x^5+14 x^4+52 x^3+9 x^2+13 x+41$
• $y^2=7 x^6+50 x^5+45 x^4+4 x^3+6 x^2+49 x+41$
• $y^2=14 x^6+41 x^5+31 x^4+8 x^3+12 x^2+39 x+23$
• $y^2=48 x^6+4 x^5+21 x^4+20 x^3+49 x^2+11 x+51$
• $y^2=37 x^6+8 x^5+42 x^4+40 x^3+39 x^2+22 x+43$
• $y^2=24 x^6+21 x^5+4 x^4+40 x^3+26 x^2+17 x+44$
• $y^2=48 x^6+42 x^5+8 x^4+21 x^3+52 x^2+34 x+29$
• $y^2=48 x^6+27 x^5+4 x^4+17 x^3+18 x^2+3 x+15$
• $y^2=37 x^6+54 x^5+8 x^4+34 x^3+36 x^2+6 x+30$
• $y^2=27 x^6+36 x^5+28 x^4+35 x^3+10 x^2+9 x+11$
• $y^2=54 x^6+13 x^5+56 x^4+11 x^3+20 x^2+18 x+22$
• $y^2=43 x^6+54 x^5+33 x^4+49 x^3+21 x^2+27 x+42$
• $y^2=27 x^6+49 x^5+7 x^4+39 x^3+42 x^2+54 x+25$

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 \(\Q(\sqrt{26}, \sqrt{-210})\).

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

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.do^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1365}) \)$)$

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_ado	$4$	(not in LMFDB)