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

• Label: 2.37.as_fp
• 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:	$1 - 18 x + 145 x^{2} - 666 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.00735490858575$, $\pm0.340688241919$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{10})\)
Galois group:	$C_2^2$
Jacobians:	$2$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$831$	$1827369$	$2565626076$	$3509515158201$	$4806748586136711$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$1336$	$50654$	$1872580$	$69317480$	$2565525742$	$94931050808$	$3512478208324$	$129961739795078$	$4808584277410936$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=x^6+x^3+20$
• $y^2=x^6+2x^3+18$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{6}}$.

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{10})\).

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

The base change of $A$ to $\F_{37^{6}}$ is 1.2565726409.afsla^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{37^{2}}$

  The base change of $A$ to $\F_{37^{2}}$ is the simple isogeny class 2.1369.abi_aif and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{10})\).

• Endomorphism algebra over $\F_{37^{3}}$

  The base change of $A$ to $\F_{37^{3}}$ is the simple isogeny class 2.50653.a_afsla and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{10})\).

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.a_bi	$3$	(not in LMFDB)
2.37.s_fp	$3$	(not in LMFDB)
2.37.a_bi	$6$	(not in LMFDB)