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

• Label: 2.31.at_fv
• Base field: $\F_{31}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{31}$
Dimension:	$2$
L-polynomial:	$1 - 19 x + 151 x^{2} - 589 x^{3} + 961 x^{4}$
Frobenius angles:	$\pm0.0974224575311$, $\pm0.228740076827$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\zeta_{5})\)
Galois group:	$C_4$
Jacobians:	$3$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$505$	$869105$	$886957255$	$854000824205$	$819891613750000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$903$	$29773$	$924723$	$28638348$	$887542563$	$27512689903$	$852890868723$	$26439621574123$	$819628308589198$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=26 x^5+28$
• $y^2=6 x^6+20 x^5+12 x^4+2 x^3+30 x^2+21 x$
• $y^2=3 x^6+22 x^4+25 x^3+18 x^2+14 x+22$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{31}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).

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.t_fv	$2$	(not in LMFDB)
2.31.ae_bu	$5$	(not in LMFDB)
2.31.b_abn	$5$	(not in LMFDB)