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

• Label: 2.31.ad_bi
• 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:	$( 1 - 7 x + 31 x^{2} )( 1 + 4 x + 31 x^{2} )$
	$1 - 3 x + 34 x^{2} - 93 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.283620691308$, $\pm0.616954024641$
Angle rank:	$1$ (numerical)
Jacobians:	$76$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$900$	$982800$	$887468400$	$854332315200$	$820001896897500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$29$	$1021$	$29792$	$925081$	$28642199$	$887433118$	$27512103809$	$852891620881$	$26439622160672$	$819628287497701$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 76 curves (of which all are hyperelliptic):

• $y^2=3 x^6+10 x^5+14 x^4+23 x^3+6 x+8$
• $y^2=11 x^6+11 x^5+4 x^3+14 x^2+4 x+24$
• $y^2=25 x^6+20 x^5+17 x^4+23 x^3+13 x^2+10 x+8$
• $y^2=22 x^6+18 x^5+30 x^4+16 x^3+9 x^2+23 x+26$
• $y^2=20 x^6+13 x^5+27 x^4+11 x^3+23 x^2+4 x+18$
• $y^2=24 x^6+x^5+19 x^4+6 x^3+29 x^2+20 x+14$
• $y^2=15 x^6+21 x^5+18 x^3+6 x^2+8 x+12$
• $y^2=12 x^6+18 x^5+23 x^4+12 x^3+29 x^2+19 x+12$
• $y^2=x^6+x^3+6$
• $y^2=19 x^6+25 x^5+24 x^4+11 x^2+29 x+24$
• $y^2=12 x^5+29 x^4+7 x^3+6 x^2+21 x+5$
• $y^2=14 x^6+4 x^5+4 x^4+4 x^3+7 x^2+9 x+13$
• $y^2=30 x^5+9 x^4+4 x^3+12 x^2+24 x+27$
• $y^2=10 x^6+6 x^5+9 x^4+2 x^3+30 x^2+9 x+11$
• $y^2=30 x^6+21 x^5+5 x^4+17 x^3+21 x^2+10 x+9$
• $y^2=30 x^6+17 x^5+4 x^4+5 x^3+8 x^2+12 x+11$
• $y^2=7 x^6+15 x^5+28 x^4+17 x^3+12 x^2+9 x+26$
• $y^2=10 x^6+x^5+16 x^4+21 x^3+17 x^2+9 x+4$
• $y^2=25 x^6+2 x^4+27 x^3+30 x^2+20 x+6$
• $y^2=17 x^6+9 x^4+5 x^3+18 x^2+26 x+4$
• and

• $y^2=6 x^6+23 x^5+29 x^4+9 x^3+12 x^2+5 x+15$
• $y^2=25 x^6+16 x^5+3 x^4+24 x^3+3 x^2+13 x$
• $y^2=26 x^6+3 x^5+12 x^4+26 x^3+2 x^2+26 x+26$
• $y^2=24 x^6+12 x^5+10 x^4+24 x^3+16 x^2+27 x$
• $y^2=28 x^6+24 x^5+21 x^4+25 x^3+12 x^2+7 x+15$
• $y^2=23 x^6+17 x^5+2 x^4+26 x^3+18 x^2+8 x+29$
• $y^2=3 x^6+3 x^5+12 x^4+27 x^3+10 x^2+21$
• $y^2=13 x^6+24 x^5+23 x^4+29 x^3+22 x^2+23 x+29$
• $y^2=12 x^6+24 x^5+30 x^4+8 x^3+3 x^2+21 x+29$
• $y^2=21 x^6+x^5+20 x^4+29 x^3+24 x^2+27 x+23$
• $y^2=16 x^6+29 x^5+3 x^4+26 x^3+5 x^2+13$
• $y^2=21 x^6+2 x^5+6 x^4+13 x^3+14 x^2+5 x+3$
• $y^2=28 x^6+5 x^5+27 x^4+7 x^3+20 x^2+24 x+29$
• $y^2=15 x^6+14 x^5+5 x^4+19 x^3+4 x^2+23 x+6$
• $y^2=22 x^6+13 x^5+20 x^4+7 x^3+6 x^2+28 x$
• $y^2=18 x^6+6 x^5+5 x^4+12 x^3+5 x^2+19 x+26$
• $y^2=12 x^5+15 x^4+13 x^2+15 x+6$
• $y^2=21 x^6+12 x^5+15 x^4+10 x^3+30 x^2+21 x+13$
• $y^2=8 x^6+18 x^5+14 x^4+18 x^3+28 x^2+7 x+17$
• $y^2=26 x^6+19 x^5+24 x^4+13 x^3+11 x^2+24 x+30$
• $y^2=26 x^6+13 x^5+26 x^4+18 x^3+30 x^2+29 x+11$
• $y^2=11 x^6+29 x^5+6 x^4+x^3+20 x^2+4 x+23$
• $y^2=9 x^6+25 x^5+16 x^4+19 x^3+16 x^2+5 x+6$
• $y^2=17 x^6+7 x^5+20 x^4+3 x^3+20 x^2+4 x+3$
• $y^2=5 x^6+23 x^5+7 x^4+2 x^3+2 x^2+16 x+4$
• $y^2=14 x^6+29 x^5+12 x^4+x^3+27 x^2+11 x+23$
• $y^2=29 x^6+20 x^5+24 x^4+4 x^3+24 x^2+10 x+28$
• $y^2=9 x^6+21 x^5+18 x^4+13 x^3+8 x^2+2 x+21$
• $y^2=10 x^6+22 x^5+8 x^4+21 x^3+11 x^2+5 x+13$
• $y^2=7 x^6+27 x^5+30 x^4+16 x^3+22 x^2+26 x+17$
• $y^2=8 x^6+12 x^5+26 x^4+4 x^3+23 x^2+17 x+3$
• $y^2=6 x^6+18 x^5+19 x^4+4 x^3+6 x^2+22 x+9$
• $y^2=3 x^6+27 x^5+12 x^4+28 x^3+19 x^2+22 x+27$
• $y^2=13 x^6+16 x^5+22 x^4+4 x^3+28 x^2+3 x+23$
• $y^2=16 x^6+23 x^5+13 x^4+18 x^3+28 x^2+22 x+6$
• $y^2=25 x^6+24 x^5+x^4+30 x^3+15 x^2+21 x+26$
• $y^2=13 x^6+19 x^5+x^4+x^3+25 x^2+x+16$
• $y^2=6 x^6+15 x^5+12 x^4+26 x^3+21 x^2+7 x+2$
• $y^2=9 x^6+27 x^5+5 x^4+2 x^3+21 x^2+21 x+1$
• $y^2=3 x^6+9 x^3+4$
• $y^2=26 x^6+24 x^5+29 x^4+12 x^3+2 x+14$
• $y^2=25 x^6+17 x^5+29 x^4+22 x^3+24 x^2+10 x$
• $y^2=13 x^6+15 x^5+6 x^4+15 x^3+2 x^2+15 x+26$
• $y^2=20 x^6+6 x^5+20 x^4+6 x^3+20 x^2+20 x+9$
• $y^2=x^6+2 x^5+13 x^4+20 x^3+27 x^2+15 x+9$
• $y^2=15 x^6+10 x^5+7 x^4+17 x^3+24 x^2+29 x+23$
• $y^2=30 x^6+9 x^5+2 x^4+3 x^3+3 x+29$
• $y^2=18 x^6+21 x^5+2 x^4+26 x^3+26 x^2+x+14$
• $y^2=17 x^6+18 x^5+12 x^4+2 x^3+12 x^2+16 x+8$
• $y^2=28 x^6+23 x^5+23 x^4+30 x^3+18 x^2+x+4$
• $y^2=17 x^6+11 x^5+4 x^4+16 x^3+25 x^2+17 x+13$
• $y^2=21 x^6+23 x^5+13 x^4+3 x^3+15 x^2+22 x+27$
• $y^2=26 x^6+16 x^5+16 x^4+20 x^3+4 x^2+29 x+27$
• $y^2=17 x^6+2 x^5+8 x^4+13 x^3+11 x^2+13 x+26$
• $y^2=30 x^6+10 x^5+21 x^4+17 x^3+28 x^2+12 x+3$
• $y^2=29 x^6+6 x^5+26 x^4+27 x^3+10 x^2+23$

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.ah $\times$ 1.31.e 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.ah : \(\Q(\sqrt{-3}) \).
• 1.31.e : \(\Q(\sqrt{-3}) \).

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.n $\times$ 1.961.bu. The endomorphism algebra for each factor is:

  • 1.961.n : \(\Q(\sqrt{-3}) \).
  • 1.961.bu : \(\Q(\sqrt{-3}) \).
• 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.al_dm	$2$	(not in LMFDB)
2.31.l_dm	$2$	(not in LMFDB)
2.31.as_fj	$3$	(not in LMFDB)
2.31.ap_ec	$3$	(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)
2.31.s_fj	$3$	(not in LMFDB)