Abelian variety isogeny class 2.83.abe_ot over $\F_{83}$

• Label: 2.83.abe_ot
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$1 - 30 x + 383 x^{2} - 2490 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.0661835162234$, $\pm0.267149817110$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$21$
Isomorphism classes:	21

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4753$	$46546129$	$326940010096$	$2252490483005721$	$15516034691728738993$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$6756$	$571788$	$47462500$	$3939038994$	$326939646822$	$27136040228358$	$2252292154678084$	$186940255267540404$	$15516041195083027236$

Jacobians and polarizations

This isogeny class contains the Jacobians of 21 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=46x^6+56x^5+3x^4+40x^3+27x^2+43x+2$
• $y^2=57x^6+56x^5+60x^4+78x^3+54x^2+37x+57$
• $y^2=35x^6+19x^5+3x^4+35x^3+40x^2+6x+60$
• $y^2=24x^6+54x^5+65x^4+30x^3+31x^2+17x+47$
• $y^2=35x^6+51x^5+61x^4+28x^3+64x^2+21x+13$
• $y^2=55x^6+19x^5+46x^4+60x^3+47x^2+61x+46$
• $y^2=60x^6+28x^5+24x^4+20x^3+54x^2+43x+61$
• $y^2=20x^6+29x^5+22x^4+32x^3+45x^2+3x+61$
• $y^2=29x^6+40x^5+72x^4+20x^3+45x^2+65x+32$
• $y^2=24x^6+47x^5+38x^4+76x^3+12x^2+48x+60$
• $y^2=54x^6+72x^5+65x^4+81x^3+14x^2+31x+5$
• $y^2=43x^6+14x^5+34x^4+x^3+49x^2+55x+66$
• $y^2=45x^6+10x^5+24x^4+5x^3+63x^2+18x+56$
• $y^2=20x^6+64x^5+26x^4+46x^3+40x^2+18x+62$
• $y^2=55x^6+9x^5+17x^4+7x^3+48x^2+78x+17$
• $y^2=4x^6+77x^5+15x^4+76x^3+41x^2+81x+43$
• $y^2=20x^6+57x^5+78x^4+71x^3+10x^2+63x+20$
• $y^2=82x^6+63x^5+9x^4+52x^3+82x^2+14x+9$
• $y^2=71x^6+63x^5+73x^4+2x^3+7x^2+20x+61$
• $y^2=6x^6+79x^5+24x^4+81x^3+45x^2+63x+22$
• $y^2=74x^6+2x^5+2x^4+70x^3+23x^2+27x+74$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83^{6}}$.

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\).

Endomorphism algebra over $\overline{\F}_{83}$

The base change of $A$ to $\F_{83^{6}}$ is 1.326940373369.aurkc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{83^{2}}$

  The base change of $A$ to $\F_{83^{2}}$ is the simple isogeny class 2.6889.afe_qjr and its endomorphism algebra is \(\Q(\sqrt{2}, \sqrt{-3})\).

• Endomorphism algebra over $\F_{83^{3}}$

  The base change of $A$ to $\F_{83^{3}}$ is the simple isogeny class 2.571787.a_aurkc and its endomorphism algebra is \(\Q(\sqrt{2}, \sqrt{-3})\).

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
2.83.be_ot	$2$	(not in LMFDB)
2.83.a_fe	$3$	(not in LMFDB)
2.83.be_ot	$3$	(not in LMFDB)