Abelian variety isogeny class 2.41.k_by over $\F_{41}$

• Label: 2.41.k_by
• Base field: $\F_{41}$
• 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:	$\F_{41}$
Dimension:	$2$
L-polynomial:	$1 + 10 x + 50 x^{2} + 410 x^{3} + 1681 x^{4}$
Frobenius angles:	$\pm0.436195342855$, $\pm0.936195342855$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{57})\)
Galois group:	$C_2^2$
Jacobians:	$32$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2152$	$2823424$	$4800683752$	$7971723083776$	$13421900473480552$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$1682$	$69652$	$2821086$	$115849652$	$4750104242$	$194755291412$	$7984925599678$	$327381890822452$	$13422659310152402$

Jacobians and polarizations

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

• $y^2=40 x^6+9 x^5+4 x^4+19 x^3+9 x^2+10 x+29$
• $y^2=30 x^6+14 x^5+10 x^4+38 x^3+32 x^2+17 x+38$
• $y^2=30 x^6+37 x^5+24 x^4+24 x^3+13 x^2+23 x+33$
• $y^2=38 x^6+7 x^5+36 x^4+4 x^3+33 x^2+27 x+21$
• $y^2=18 x^6+21 x^5+28 x^4+8 x^3+27 x^2+32 x+20$
• $y^2=10 x^6+27 x^5+28 x^4+31 x^3+6 x^2+16 x+40$
• $y^2=x^6+11 x^5+37 x^4+33 x^3+15 x^2+33 x+28$
• $y^2=26 x^5+15 x^4+6 x^3+24 x^2+22 x+14$
• $y^2=38 x^6+21 x^5+35 x^4+33 x^3+38 x^2+3 x+19$
• $y^2=26 x^6+28 x^5+4 x^4+3 x^3+29 x^2+29 x+36$
• $y^2=20 x^5+23 x^4+20 x^3+37 x^2+7 x+33$
• $y^2=26 x^6+17 x^5+19 x^4+3 x^3+28 x^2+24 x+7$
• $y^2=9 x^6+30 x^5+24 x^4+28 x^3+15 x+9$
• $y^2=38 x^6+9 x^5+11 x^4+27 x^3+40 x^2+25 x+40$
• $y^2=14 x^6+31 x^5+27 x^4+27 x^3+25 x^2+13 x+24$
• $y^2=32 x^6+25 x^5+2 x^4+4 x^3+22 x^2+35 x+6$
• $y^2=40 x^6+30 x^5+12 x^4+12 x^2+11 x+40$
• $y^2=29 x^6+34 x^5+35 x^4+24 x^3+35 x^2+40 x+32$
• $y^2=2 x^6+8 x^5+8 x^4+7 x^3+19 x^2+39 x+16$
• $y^2=31 x^6+9 x^5+33 x^4+6 x^3+28 x^2+16 x+40$
• and

• $y^2=8 x^6+36 x^5+5 x^4+36 x^3+4 x^2+28 x$
• $y^2=20 x^6+26 x^5+39 x^4+4 x^3+6 x^2+10 x+29$
• $y^2=32 x^6+4 x^5+12 x^4+12 x^2+37 x+32$
• $y^2=23 x^5+23 x^4+17 x^3+11 x^2+4 x+21$
• $y^2=2 x^6+12 x^5+31 x^4+25 x^3+37 x^2+19 x+4$
• $y^2=36 x^6+15 x^5+11 x^4+38 x^3+29 x^2+x+24$
• $y^2=7 x^6+14 x^5+12 x^4+38 x^3+26 x^2+8 x+1$
• $y^2=34 x^6+3 x^5+26 x^4+9 x^3+x^2+28 x+16$
• $y^2=29 x^6+5 x^5+4 x^4+15 x^3+28 x^2+31 x+20$
• $y^2=19 x^6+27 x^5+29 x^4+25 x^3+7 x^2+26 x$
• $y^2=25 x^6+7 x^5+37 x^4+8 x^3+26 x^2+23 x+1$
• $y^2=15 x^6+33 x^5+13 x^4+37 x^3+35 x^2+6 x+13$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{41^{4}}$.

Endomorphism algebra over $\F_{41}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{57})\).

Endomorphism algebra over $\overline{\F}_{41}$

The base change of $A$ to $\F_{41^{4}}$ is 1.2825761.adly^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{41^{2}}$

  The base change of $A$ to $\F_{41^{2}}$ is the simple isogeny class 2.1681.a_adly and its endomorphism algebra is \(\Q(i, \sqrt{57})\).

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
2.41.ak_by	$2$	(not in LMFDB)
2.41.ak_by	$4$	(not in LMFDB)
2.41.a_abg	$8$	(not in LMFDB)