Abelian variety isogeny class 2.61.e_abt over $\F_{61}$

• Label: 2.61.e_abt
• Base field: $\F_{61}$
• 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:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$1 + 4 x - 45 x^{2} + 244 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.249095665427$, $\pm0.915762332094$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{19})\)
Galois group:	$C_2^2$
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3925$	$13458825$	$51824522500$	$191759844670425$	$713390147149373125$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$66$	$3616$	$228318$	$13849636$	$844652226$	$51520389838$	$3142739393706$	$191707299700036$	$11694145779249558$	$713342913101183776$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=58 x^6+43 x^5+30 x^4+25 x^3+32 x^2+7 x+60$
• $y^2=24 x^6+51 x^5+20 x^4+46 x^3+17 x^2+17 x+9$
• $y^2=44 x^6+7 x^5+4 x^4+23 x^3+37 x^2+17 x+22$
• $y^2=2 x^6+47 x^5+49 x^4+35 x^3+20 x^2+15 x+24$
• $y^2=2 x^6+2 x^3+30$
• $y^2=x^6+44 x^5+6 x^4+38 x^3+44 x^2+23 x+10$
• $y^2=24 x^6+5 x^5+21 x^4+12 x^3+37 x^2+42 x+43$
• $y^2=13 x^6+8 x^5+41 x^4+40 x^3+56 x^2+45 x+3$
• $y^2=7 x^6+52 x^5+23 x^4+44 x^3+38 x^2+22 x+59$
• $y^2=15 x^6+20 x^5+11 x^4+38 x^3+30 x^2+30 x+27$
• $y^2=30 x^6+36 x^5+12 x^4+2 x^3+46 x^2+48 x+28$
• $y^2=20 x^6+31 x^5+13 x^4+39 x^3+13 x^2+45 x+20$
• $y^2=49 x^6+29 x^5+44 x^4+47 x^3+7 x^2+52 x+26$
• $y^2=55 x^6+12 x^5+19 x^4+x^3+17 x^2+9 x+41$
• $y^2=25 x^6+21 x^5+32 x^4+59 x^3+53 x^2+3 x+50$
• $y^2=17 x^6+32 x^5+26 x^4+47 x^3+37 x^2+35 x+27$
• $y^2=40 x^6+38 x^4+44 x^3+43 x^2+45 x+41$
• $y^2=2 x^6+47 x^5+47 x^4+35 x^3+7 x^2+x+47$
• $y^2=34 x^6+49 x^5+47 x^4+20 x^3+39 x^2+26 x+25$
• $y^2=5 x^6+54 x^5+43 x^4+9 x^3+47 x^2+37 x+27$
• and

• $y^2=x^6+32 x^5+54 x^4+8 x^3+2 x^2+21 x+27$
• $y^2=19 x^6+17 x^5+30 x^4+33 x^3+12 x^2+21 x+45$
• $y^2=45 x^6+10 x^5+43 x^4+9 x^3+48 x^2+30 x+48$
• $y^2=2 x^6+4 x^3+11$
• $y^2=36 x^6+52 x^5+5 x^4+57 x^3+9 x^2+11 x+3$
• $y^2=19 x^6+39 x^5+54 x^4+41 x^3+34 x^2+38 x+60$
• $y^2=50 x^6+27 x^5+20 x^4+32 x^3+50 x^2+48 x+19$
• $y^2=14 x^6+45 x^5+54 x^4+12 x^2+40 x+59$
• $y^2=46 x^6+18 x^5+49 x^4+55 x^3+40 x^2+45 x+33$
• $y^2=11 x^6+33 x^5+52 x^4+13 x^3+3 x^2+25 x+34$
• $y^2=58 x^6+52 x^5+29 x^4+44 x^3+7 x^2+19 x+31$
• $y^2=7 x^6+21 x^5+11 x^4+44 x^3+39 x^2+19 x+13$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{3}}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{19})\).

Endomorphism algebra over $\overline{\F}_{61}$

The base change of $A$ to $\F_{61^{3}}$ is 1.226981.zs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

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
2.61.ae_abt	$2$	(not in LMFDB)
2.61.ai_fi	$3$	(not in LMFDB)
2.61.a_ec	$6$	(not in LMFDB)