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

• Label: 2.59.d_eo
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$( 1 + 59 x^{2} )( 1 + 3 x + 59 x^{2} )$
	$1 + 3 x + 118 x^{2} + 177 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.562562653022$
Angle rank:	$1$ (numerical)
Jacobians:	$60$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3780$	$12927600$	$42077432880$	$146686529707200$	$511148563857369900$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$3709$	$204876$	$12105481$	$714968793$	$42181101142$	$2488648389507$	$146830413407761$	$8662996001163924$	$511116754180711549$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

• $y^2=4 x^6+38 x^5+33 x^4+25 x^3+6 x^2+21 x+52$
• $y^2=41 x^6+31 x^5+19 x^4+45 x^3+48 x^2+2 x+34$
• $y^2=39 x^6+43 x^5+14 x^4+50 x^3+43 x^2+41 x+27$
• $y^2=20 x^6+14 x^5+15 x^3+48 x^2+37 x+1$
• $y^2=29 x^6+54 x^5+3 x^4+24 x^3+24 x^2+49 x+24$
• $y^2=17 x^6+46 x^5+57 x^4+6 x^3+52 x^2+36 x+2$
• $y^2=24 x^6+42 x^5+51 x^4+49 x^3+24 x^2+7 x+10$
• $y^2=44 x^6+7 x^5+41 x^4+44 x^3+50 x^2+30 x+7$
• $y^2=18 x^6+24 x^5+4 x^4+12 x^3+30 x^2+34 x+1$
• $y^2=27 x^6+4 x^5+5 x^4+11 x^3+39 x^2+50 x+6$
• $y^2=36 x^6+28 x^5+56 x^4+37 x^3+12 x^2+23 x+15$
• $y^2=42 x^6+22 x^5+36 x^4+50 x^3+5 x^2+29 x+3$
• $y^2=2 x^6+13 x^5+48 x^4+x^3+28 x^2+15 x+3$
• $y^2=3 x^6+39 x^5+46 x^4+15 x^3+35 x^2+13 x+16$
• $y^2=10 x^6+56 x^4+56 x^3+41 x^2+24 x+41$
• $y^2=18 x^6+x^5+10 x^4+38 x^3+11 x^2+31 x+8$
• $y^2=28 x^6+18 x^5+36 x^4+49 x^3+5 x^2+11 x+11$
• $y^2=42 x^6+42 x^5+17 x^4+31 x^3+50 x^2+51 x+26$
• $y^2=40 x^6+50 x^5+32 x^4+11 x^3+48 x^2+53 x+15$
• $y^2=31 x^6+21 x^5+17 x^4+7 x^3+45 x^2+53 x+27$
• and

• $y^2=56 x^6+49 x^5+8 x^4+20 x^3+12 x^2+18 x$
• $y^2=42 x^6+25 x^5+53 x^4+51 x^3+28 x^2+12 x+50$
• $y^2=35 x^6+4 x^5+5 x^4+10 x^3+11 x^2+18 x+55$
• $y^2=54 x^6+20 x^5+34 x^4+10 x^3+47 x^2+16 x+45$
• $y^2=57 x^6+41 x^5+39 x^4+27 x^3+17 x^2+20 x+6$
• $y^2=30 x^6+57 x^4+44 x^3+31 x^2+11 x+3$
• $y^2=48 x^6+9 x^5+39 x^4+8 x^3+40 x^2+51 x+46$
• $y^2=8 x^6+46 x^5+26 x^4+14 x^3+39 x^2+29 x+53$
• $y^2=10 x^6+2 x^5+27 x^4+49 x^3+39 x^2+26$
• $y^2=31 x^6+24 x^5+22 x^4+43 x^3+52 x^2+14 x+42$
• $y^2=15 x^6+41 x^5+30 x^4+33 x^3+x^2+13 x+16$
• $y^2=4 x^6+25 x^5+6 x^4+46 x^3+37 x^2+47 x+52$
• $y^2=32 x^6+29 x^5+50 x^4+26 x^3+24 x^2+30 x+27$
• $y^2=3 x^6+24 x^5+10 x^4+48 x^3+51 x^2+37 x+5$
• $y^2=25 x^6+27 x^5+9 x^4+51 x^3+x^2+39 x+7$
• $y^2=34 x^6+5 x^5+12 x^4+43 x^3+18 x^2+38 x+31$
• $y^2=52 x^6+4 x^5+12 x^4+11 x^3+8 x^2+57 x+27$
• $y^2=17 x^6+15 x^5+41 x^4+4 x^3+24 x+27$
• $y^2=3 x^6+23 x^5+32 x^4+34 x^3+48 x^2+8 x+4$
• $y^2=53 x^6+54 x^5+51 x^4+17 x^3+19 x^2+3 x+28$
• $y^2=19 x^6+x^5+14 x^4+14 x^3+28 x^2+18 x+52$
• $y^2=12 x^6+43 x^5+x^4+33 x^3+47 x^2+24$
• $y^2=58 x^6+35 x^5+20 x^4+44 x^3+57 x^2+30 x+26$
• $y^2=26 x^6+55 x^5+14 x^4+45 x^3+4 x^2+7 x+24$
• $y^2=49 x^6+5 x^5+8 x^4+37 x^3+48 x^2+57 x+3$
• $y^2=4 x^6+9 x^5+28 x^4+11 x^3+48 x^2+17 x+45$
• $y^2=9 x^6+20 x^5+12 x^4+33 x^3+13 x^2+20 x+57$
• $y^2=14 x^6+37 x^5+56 x^4+45 x^3+9 x^2+21 x+50$
• $y^2=26 x^6+29 x^5+26 x^4+8 x^3+41 x^2+45 x+34$
• $y^2=26 x^6+4 x^5+15 x^4+51 x^3+53 x^2+30 x+11$
• $y^2=42 x^6+48 x^5+24 x^4+45 x^3+28 x^2+47 x+12$
• $y^2=26 x^6+55 x^5+55 x^4+33 x^3+30 x^2+10 x+17$
• $y^2=10 x^6+x^5+34 x^4+36 x^3+13 x^2+43 x+16$
• $y^2=53 x^6+38 x^5+54 x^4+7 x^3+21 x^2+5 x+41$
• $y^2=24 x^6+34 x^5+19 x^4+54 x^3+35 x^2+39 x+23$
• $y^2=39 x^6+54 x^5+32 x^4+56 x^3+11 x^2+55 x+52$
• $y^2=51 x^6+39 x^5+24 x^4+50 x^3+4 x^2+34 x+3$
• $y^2=6 x^6+21 x^5+3 x^4+24 x^3+47 x^2+3 x+30$
• $y^2=42 x^6+40 x^5+6 x^4+33 x^3+13 x^2+44 x+31$
• $y^2=18 x^6+47 x^5+5 x^4+8 x^3+6 x^2+18 x+54$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.a $\times$ 1.59.d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.59.a : \(\Q(\sqrt{-59}) \).
• 1.59.d : \(\Q(\sqrt{-227}) \).

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

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.ef $\times$ 1.3481.eo. The endomorphism algebra for each factor is:

• 1.3481.ef : \(\Q(\sqrt{-227}) \).
• 1.3481.eo : the quaternion algebra over \(\Q\) ramified at $59$ and $\infty$.

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.ad_eo	$2$	(not in LMFDB)