Abelian variety isogeny class 2.37.b_abk over $\F_{37}$

• Label: 2.37.b_abk
• Base field: $\F_{37}$
• 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:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 37 x^{2} )( 1 + 11 x + 37 x^{2} )$
	$1 + x - 36 x^{2} + 37 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.192861133077$, $\pm0.859527799744$
Angle rank:	$1$ (numerical)
Jacobians:	$35$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1372$	$1778112$	$2576983696$	$3517340246784$	$4809046247547052$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$39$	$1297$	$50874$	$1876753$	$69350619$	$2565904822$	$94931541471$	$3512482418881$	$129961709026098$	$4808584278098857$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 35 curves (of which all are hyperelliptic):

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

• $y^2=34 x^6+7 x^5+21 x^4+9 x^3+36 x^2+11 x+23$
• $y^2=15 x^6+22 x^5+7 x^4+23 x^3+14 x^2+25 x+34$
• $y^2=24 x^6+31 x^5+25 x^4+32 x^3+15 x^2+18 x+32$
• $y^2=7 x^6+26 x^5+32 x^4+7 x^3+16 x^2+35 x+32$
• $y^2=30 x^6+4 x^4+15 x^3+4 x^2+15 x+19$
• $y^2=36 x^6+34 x^5+36 x^4+5 x^3+36 x^2+24 x+10$
• $y^2=14 x^6+9 x^5+3 x^4+14 x^3+16 x^2+18 x$
• $y^2=21 x^6+20 x^5+3 x^4+27 x^3+14 x^2+22 x+7$
• $y^2=20 x^6+3 x^5+32 x^4+9 x^3+16 x^2+14$
• $y^2=4 x^6+28 x^5+30 x^4+3 x^3+28 x^2+22 x+1$
• $y^2=35 x^6+29 x^5+7 x^4+21 x^3+35 x^2+33 x+10$
• $y^2=35 x^6+32 x^5+29 x^4+25 x^3+35 x^2+14 x+35$
• $y^2=36 x^6+13 x^5+4 x^4+14 x^3+6 x^2+9 x+36$
• $y^2=25 x^6+17 x^5+16 x^4+x^3+15 x^2+2 x+19$
• $y^2=13 x^6+34 x^5+23 x^4+25 x^3+13 x^2+15 x+25$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{3}}$.

Endomorphism algebra over $\F_{37}$

The isogeny class factors as 1.37.ak $\times$ 1.37.l and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.37.ak : \(\Q(\sqrt{-3}) \).
• 1.37.l : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{37}$

The base change of $A$ to $\F_{37^{3}}$ is 1.50653.eg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
2.37.av_hc	$2$	(not in LMFDB)
2.37.ab_abk	$2$	(not in LMFDB)
2.37.v_hc	$2$	(not in LMFDB)
2.37.au_gs	$3$	(not in LMFDB)
2.37.al_dg	$3$	(not in LMFDB)
2.37.ac_cx	$3$	(not in LMFDB)
2.37.k_cl	$3$	(not in LMFDB)
2.37.w_hn	$3$	(not in LMFDB)