Abelian variety isogeny class 2.31.s_fj over $\F_{31}$

• Label: 2.31.s_fj
• Base field: $\F_{31}$
• 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:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$( 1 + 7 x + 31 x^{2} )( 1 + 11 x + 31 x^{2} )$
	$1 + 18 x + 139 x^{2} + 558 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.716379308692$, $\pm0.950287357974$
Angle rank:	$1$ (numerical)
Jacobians:	$6$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1677$	$880425$	$887468400$	$853069314825$	$819767976545037$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$916$	$29792$	$923716$	$28634030$	$887433118$	$27513097970$	$852889228036$	$26439622160672$	$819628336307476$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=x^6+x^3+26$
• $y^2=13 x^6+16 x^5+29 x^4+14 x^3+3 x^2+15 x+7$
• $y^2=8 x^6+8 x^5+23 x^4+23 x^3+23 x^2+8 x+8$
• $y^2=2 x^6+23 x^5+22 x^4+30 x^3+4 x^2+3 x+30$
• $y^2=3 x^6+4$
• $y^2=28 x^6+25 x^5+21 x^4+10 x^3+21 x^2+25 x+28$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{6}}$.

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.h $\times$ 1.31.l and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.31.h : \(\Q(\sqrt{-3}) \).
• 1.31.l : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{31}$

The base change of $A$ to $\F_{31^{6}}$ is 1.887503681.acafa^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{31^{2}}$
  The base change of $A$ to $\F_{31^{2}}$ is 1.961.ach $\times$ 1.961.n. The endomorphism algebra for each factor is:

  • 1.961.ach : \(\Q(\sqrt{-3}) \).
  • 1.961.n : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{31^{3}}$
  The base change of $A$ to $\F_{31^{3}}$ is 1.29791.alw $\times$ 1.29791.lw. The endomorphism algebra for each factor is:

  • 1.29791.alw : \(\Q(\sqrt{-3}) \).
  • 1.29791.lw : \(\Q(\sqrt{-3}) \).

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
2.31.ae_ap	$2$	(not in LMFDB)
2.31.e_ap	$2$	(not in LMFDB)
2.31.as_fj	$3$	(not in LMFDB)
2.31.ap_ec	$3$	(not in LMFDB)
2.31.ad_bi	$3$	(not in LMFDB)
2.31.a_ach	$3$	(not in LMFDB)
2.31.a_n	$3$	(not in LMFDB)
2.31.a_bu	$3$	(not in LMFDB)
2.31.d_bi	$3$	(not in LMFDB)
2.31.p_ec	$3$	(not in LMFDB)