Abelian variety isogeny class 2.43.ae_dm over $\F_{43}$

• Label: 2.43.ae_dm
• Base field: $\F_{43}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x + 43 x^{2} )^{2}$
	$1 - 4 x + 90 x^{2} - 172 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.451268054243$, $\pm0.451268054243$
Angle rank:	$1$ (numerical)
Jacobians:	$32$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 7$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1764$	$3732624$	$6361338564$	$11667525682176$	$21606542817856164$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$2014$	$80008$	$3412750$	$146974840$	$6321556078$	$271820442136$	$11688195639454$	$502592523926344$	$21611482336703614$

Jacobians and polarizations

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

• $y^2=29 x^6+36 x^5+35 x^4+2 x^3+24 x^2+31 x+42$
• $y^2=10 x^6+13 x^5+14 x^4+16 x^3+34 x^2+31 x+26$
• $y^2=16 x^6+10 x^5+35 x^4+13 x^3+36 x^2+37$
• $y^2=18 x^6+27 x^5+30 x^4+12 x^3+20 x^2+34 x+4$
• $y^2=14 x^6+19 x^5+4 x^4+38 x^3+35 x^2+33 x+17$
• $y^2=14 x^6+7 x^5+30 x^4+41 x^3+30 x^2+7 x+14$
• $y^2=37 x^6+40 x^5+28 x^4+30 x^3+28 x^2+40 x+37$
• $y^2=25 x^6+16 x^5+21 x^4+19 x^3+15 x^2+38 x+14$
• $y^2=20 x^6+33 x^5+42 x^4+23 x^3+33 x^2+24 x+41$
• $y^2=24 x^6+15 x^5+31 x^4+31 x^3+16 x^2+30 x+5$
• $y^2=34 x^6+10 x^4+10 x^2+34$
• $y^2=34 x^6+38 x^5+15 x^4+24 x^3+12 x^2+8 x+14$
• $y^2=11 x^6+9 x^5+33 x^4+28 x^3+9 x^2+28 x+18$
• $y^2=21 x^6+17 x^5+20 x^4+35 x^3+20 x^2+17 x+21$
• $y^2=10 x^5+16 x^3+33 x^2+9 x+31$
• $y^2=34 x^6+41 x^5+9 x^4+9 x^3+9 x^2+41 x+34$
• $y^2=x^6+36 x^3+1$
• $y^2=10 x^6+8 x^5+6 x^4+28 x^3+6 x^2+8 x+10$
• $y^2=13 x^6+34 x^5+25 x^4+16 x^3+42 x^2+12 x+38$
• $y^2=37 x^6+36 x^5+40 x^4+40 x^3+9 x^2+23 x+33$
• and

• $y^2=29 x^6+35 x^4+35 x^2+29$
• $y^2=28 x^6+12 x^5+5 x^4+31 x^3+7 x^2+41 x+32$
• $y^2=18 x^6+20 x^5+28 x^4+2 x^3+5 x^2+28 x+15$
• $y^2=22 x^6+3 x^5+23 x^4+42 x^3+6 x^2+5 x+32$
• $y^2=42 x^6+6 x^5+28 x^4+21 x^3+26 x^2+36 x+5$
• $y^2=7 x^6+17 x^5+41 x^4+3 x^3+24 x^2+41 x+24$
• $y^2=22 x^6+24 x^5+20 x^4+14 x^3+35 x^2+40 x+12$
• $y^2=10 x^6+20 x^5+28 x^4+8 x^3+40 x^2+31 x+34$
• $y^2=5 x^6+30 x^5+31 x^4+25 x^3+36 x^2+4 x+29$
• $y^2=19 x^6+15 x^5+10 x^4+25 x^3+33 x^2+10 x+36$
• $y^2=x^6+25 x^3+16$
• $y^2=32 x^6+36 x^5+17 x^4+40 x^3+17 x^2+36 x+32$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43}$.

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$

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
2.43.a_de	$2$	(not in LMFDB)
2.43.e_dm	$2$	(not in LMFDB)
2.43.c_abn	$3$	(not in LMFDB)