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

• Label: 2.37.c_abh
• Base field: $\F_{37}$
• 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:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$1 + 2 x - 33 x^{2} + 74 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.219235123378$, $\pm0.885901790045$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$18$
Isomorphism classes:	74
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1413$	$1781793$	$2587553424$	$3516534192249$	$4809433361674293$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$1300$	$51082$	$1876324$	$69356200$	$2565837430$	$94931313160$	$3512480379844$	$129961694357314$	$4808584383596500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

• $y^2=x^6+x^3+25$
• $y^2=2 x^6+25 x^5+16 x^4+20 x^3+2 x^2+22 x+8$
• $y^2=x^6+31 x^5+36 x^4+30 x^3+27 x^2+17 x+34$
• $y^2=27 x^6+36 x^5+7 x^4+6 x^3+36 x^2+28 x+36$
• $y^2=2 x^6+x^5+29 x^4+6 x^3+25 x^2+3 x+34$
• $y^2=17 x^6+30 x^5+x^4+9 x^3+13 x^2+29 x+1$
• $y^2=x^6+x^3+34$
• $y^2=x^6+2 x^3+4$
• $y^2=17 x^6+33 x^5+10 x^4+13 x^3+9 x^2+5 x+12$
• $y^2=31 x^6+8 x^5+25 x^4+18 x^3+9 x^2+9 x+14$
• $y^2=x^6+2 x^3+33$
• $y^2=23 x^6+29 x^5+24 x^4+17 x^3+5 x^2+11 x+27$
• $y^2=24 x^6+33 x^5+4 x^4+2 x^3+6 x^2+6 x+5$
• $y^2=x^6+x^3+33$
• $y^2=4 x^6+28 x^5+19 x^4+27 x^3+4 x^2+x+19$
• $y^2=27 x^6+19 x^5+29 x^4+20 x^3+31 x^2+30 x+26$
• $y^2=24 x^6+23 x^5+11 x^4+5 x^3+x^2+35 x+14$
• $y^2=x^6+2 x^3+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{37^{3}}$ is 1.50653.ig^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.ac_abh	$2$	(not in LMFDB)
2.37.ae_da	$3$	(not in LMFDB)
2.37.am_ed	$4$	(not in LMFDB)