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

• Label: 2.61.d_aca
• 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:	$1 + 3 x - 52 x^{2} + 183 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.228181970425$, $\pm0.894848637092$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-235})\)
Galois group:	$C_2^2$
Jacobians:	$26$
Isomorphism classes:	88

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3856$	$13434304$	$51758070016$	$191781084328704$	$713383304234276176$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$65$	$3609$	$228026$	$13851169$	$844644125$	$51520737318$	$3142739374385$	$191707313682529$	$11694145666402946$	$713342912260729329$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

• $y^2=22 x^6+43 x^5+18 x^4+31 x^3+43 x^2+45 x+15$
• $y^2=35 x^5+57 x^4+41 x^3+41 x^2+50 x+59$
• $y^2=57 x^6+35 x^5+53 x^4+47 x^3+55 x^2+2 x+32$
• $y^2=20 x^6+16 x^4+51 x^3+12 x^2+15 x+55$
• $y^2=11 x^6+6 x^5+42 x^4+10 x^3+28 x^2+40 x+18$
• $y^2=56 x^6+2 x^5+30 x^4+11 x^3+21 x^2+41 x+17$
• $y^2=34 x^6+17 x^5+35 x^4+37 x^3+43 x^2+47 x+28$
• $y^2=39 x^6+27 x^5+40 x^4+58 x^3+5 x^2+16 x+19$
• $y^2=27 x^6+46 x^5+30 x^4+9 x^3+20 x^2+14 x+28$
• $y^2=35 x^6+48 x^5+33 x^4+59 x^3+26 x^2+44 x+45$
• $y^2=5 x^6+22 x^5+2 x^4+39 x^3+8 x^2+22 x+24$
• $y^2=55 x^6+42 x^5+5 x^4+33 x^3+24 x^2+52 x+13$
• $y^2=23 x^6+x^5+12 x^4+x^3+13 x^2+9 x+16$
• $y^2=7 x^6+56 x^5+45 x^4+32 x^3+15 x^2+43 x+41$
• $y^2=8 x^6+7 x^5+45 x^4+2 x^3+39 x^2+49 x+26$
• $y^2=48 x^6+48 x^5+31 x^4+20 x^3+12 x^2+57 x+36$
• $y^2=51 x^5+24 x^4+10 x^3+41 x^2+27 x+32$
• $y^2=8 x^6+56 x^5+20 x^4+57 x^3+14 x^2+35 x+45$
• $y^2=35 x^6+59 x^5+46 x^4+54 x^3+55 x^2+48 x+24$
• $y^2=24 x^6+30 x^5+6 x^4+31 x^3+40 x^2+41 x+13$
• $y^2=30 x^6+23 x^5+48 x^4+22 x^3+51 x^2+x+8$
• $y^2=52 x^6+35 x^4+52 x^3+58 x^2+9 x+60$
• $y^2=37 x^6+x^5+21 x^4+4 x^3+26 x^2+60 x+35$
• $y^2=9 x^6+31 x^5+33 x^4+2 x^3+46 x^2+18 x+7$
• $y^2=49 x^6+14 x^5+8 x^4+57 x^3+43 x^2+49 x+15$
• $y^2=2 x^6+42 x^5+x^4+31 x^3+4 x^2+5 x+37$

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 \(\Q(\sqrt{-3}, \sqrt{-235})\).

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

The base change of $A$ to $\F_{61^{3}}$ is 1.226981.uc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-235}) \)$)$

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.ad_aca	$2$	(not in LMFDB)
2.61.ag_fb	$3$	(not in LMFDB)
2.61.a_ej	$6$	(not in LMFDB)