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

• Label: 2.31.ae_ap
• 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:	$( 1 - 11 x + 31 x^{2} )( 1 + 7 x + 31 x^{2} )$
	$1 - 4 x - 15 x^{2} - 124 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.0497126420257$, $\pm0.716379308692$
Angle rank:	$1$ (numerical)
Jacobians:	$11$
Isomorphism classes:	144
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$819$	$880425$	$869306256$	$853069314825$	$819332797571379$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$916$	$29176$	$923716$	$28618828$	$887433118$	$27512793028$	$852889228036$	$26439618778216$	$819628336307476$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):

• $y^2=5 x^6+26 x^5+13 x^4+6 x^3+x^2+16 x+19$
• $y^2=18 x^6+29 x^5+30 x^4+18 x^3+22 x^2+14 x+10$
• $y^2=x^6+x^3+28$
• $y^2=3 x^6+5 x^5+24 x^4+27 x^3+12 x^2+15 x+4$
• $y^2=8 x^6+9 x^5+8 x^4+10 x^3+18 x^2+18 x+16$
• $y^2=4 x^6+5 x^5+12 x^4+11 x^3+30 x^2+8 x+16$
• $y^2=x^6+x^3+9$
• $y^2=x^6+3 x^3+18$
• $y^2=x^6+28$
• $y^2=6 x^6+10 x^5+4 x^4+6 x^3+10 x^2+16 x+24$
• $y^2=x^6+x^3+20$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{31^{3}}$.

Endomorphism algebra over $\F_{31}$

The isogeny class factors as 1.31.al $\times$ 1.31.h 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.al : \(\Q(\sqrt{-3}) \).
• 1.31.h : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{31^{3}}$ is 1.29791.alw^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
2.31.as_fj	$2$	(not in LMFDB)
2.31.e_ap	$2$	(not in LMFDB)
2.31.s_fj	$2$	(not in LMFDB)
2.31.aw_hb	$3$	(not in LMFDB)
2.31.ah_s	$3$	(not in LMFDB)
2.31.i_da	$3$	(not in LMFDB)
2.31.l_dm	$3$	(not in LMFDB)
2.31.o_eh	$3$	(not in LMFDB)