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

• Label: 2.61.a_aeo
• 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:	$1 - 118 x^{2} + 3721 x^{4}$
Frobenius angles:	$\pm0.0408675878105$, $\pm0.959132412189$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{15})\)
Galois group:	$C_2^2$
Jacobians:	$12$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3604$	$12988816$	$51520048564$	$191527885209600$	$713342911184869204$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$3486$	$226982$	$13832878$	$844596302$	$51519722766$	$3142742836022$	$191707284347998$	$11694146092834142$	$713342910706855806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=50 x^6+28 x^5+40 x^4+7 x^2+24 x+23$
• $y^2=6 x^6+2 x^5+19 x^4+3 x^2+28 x+6$
• $y^2=26 x^6+35 x^5+40 x+44$
• $y^2=11 x^6+49 x^5+3 x^4+5 x^2+13 x+8$
• $y^2=42 x^6+30 x^5+45 x^4+9 x^2+11 x+14$
• $y^2=29 x^6+7 x^5+52 x^4+19 x^2+17 x+17$
• $y^2=6 x^6+19 x^5+34 x^4+11 x^3+32 x^2+x+46$
• $y^2=49 x^6+6 x^5+34 x^4+19 x^3+37 x^2+7 x+35$
• $y^2=59 x^6+50 x^5+44 x^4+33 x^2+10 x+43$
• $y^2=9 x^6+29 x^5+7 x^4+30 x^2+40 x+9$
• $y^2=59 x^6+57 x^5+9 x^4+13 x^3+53 x^2+36 x+45$
• $y^2=17 x^6+31 x^5+33 x^4+20 x^3+42 x^2+24 x+57$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{2}}$.

Endomorphism algebra over $\F_{61}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{15})\).

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

The base change of $A$ to $\F_{61^{2}}$ is 1.3721.aeo^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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_ew	$4$	(not in LMFDB)
2.61.a_eo	$4$	(not in LMFDB)
2.61.e_ew	$4$	(not in LMFDB)