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

• Label: 2.37.ad_abc
• 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:	$1 - 3 x - 28 x^{2} - 111 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.0873537851106$, $\pm0.754020451777$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-139})\)
Galois group:	$C_2^2$
Jacobians:	$21$
Isomorphism classes:	39

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1228$	$1787968$	$2534921104$	$3515266669824$	$4807490096487868$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$1305$	$50042$	$1875649$	$69328175$	$2565741750$	$94932484115$	$3512477916769$	$129961775488754$	$4808584482833025$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{-139})\).

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

The base change of $A$ to $\F_{37^{3}}$ is 1.50653.alu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-139}) \)$)$

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.d_abc	$2$	(not in LMFDB)
2.37.g_df	$3$	(not in LMFDB)
2.37.ag_df	$6$	(not in LMFDB)