Abelian variety isogeny class 2.173.abt_bgq over $\F_{173}$

• Label: 2.173.abt_bgq
• Base field: $\F_{173}$
• 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:	$\F_{173}$
Dimension:	$2$
L-polynomial:	$1 - 45 x + 848 x^{2} - 7785 x^{3} + 29929 x^{4}$
Frobenius angles:	$\pm0.116569177593$, $\pm0.216764155741$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:	$C_2^2$
Jacobians:	$44$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$22948$	$885976384$	$26808759403456$	$802402518753628416$	$24013957048436101180468$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$129$	$29601$	$5177718$	$895793425$	$154964854869$	$26808765474822$	$4637914433102793$	$802359179025516289$	$138808137876363860814$	$24013807852611897137961$

Jacobians and polarizations

This isogeny class contains the Jacobians of 44 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=72x^6+166x^5+11x^3+35x^2+112x+165$
• $y^2=107x^6+93x^5+40x^4+106x^3+87x^2+113x+105$
• $y^2=72x^6+52x^5+132x^4+69x^3+158x^2+172x+86$
• $y^2=48x^6+31x^5+54x^4+124x^3+14x^2+105x+108$
• $y^2=131x^6+15x^5+29x^4+128x^3+38x^2+147x+30$
• $y^2=18x^6+74x^5+91x^4+172x^3+159x^2+48x+124$
• $y^2=75x^6+27x^5+65x^4+148x^3+55x^2+131x+94$
• $y^2=156x^6+144x^5+76x^4+162x^3+60x+162$
• $y^2=30x^6+146x^5+170x^4+50x^3+57x^2+73x+124$
• $y^2=26x^6+46x^5+24x^4+16x^3+138x^2+129x+134$
• $y^2=107x^6+25x^5+102x^4+5x^3+97x^2+170x+154$
• $y^2=71x^6+15x^5+170x^4+110x^3+122x^2+65x+71$
• $y^2=79x^6+167x^5+100x^4+77x^3+39x^2+161x+52$
• $y^2=68x^6+8x^5+161x^4+27x^3+86x^2+89x+55$
• $y^2=46x^6+31x^5+172x^4+170x^2+26x+43$
• $y^2=71x^6+82x^5+31x^4+87x^3+112x^2+118x+10$
• $y^2=147x^6+47x^5+164x^4+42x^3+150x^2+96x+153$
• $y^2=144x^6+77x^5+125x^4+102x^3+18x^2+24x+72$
• $y^2=43x^6+x^5+21x^4+152x^3+86x+2$
• $y^2=36x^6+14x^5+23x^4+84x^3+145x^2+155x+128$
• and

• $y^2=8x^6+81x^5+82x^4+116x^3+139x^2+163x+32$
• $y^2=123x^6+15x^5+150x^4+9x^3+18x^2+19x+56$
• $y^2=19x^6+x^5+64x^4+40x^3+91x^2+105x+129$
• $y^2=127x^6+72x^5+172x^4+144x^3+75x^2+72x+98$
• $y^2=120x^6+118x^5+165x^4+29x^3+124x^2+148x+32$
• $y^2=76x^6+141x^5+99x^4+70x^3+67x^2+141x+77$
• $y^2=146x^6+65x^5+90x^4+84x^3+91x^2+153x+33$
• $y^2=39x^6+60x^5+101x^4+169x^3+82x^2+38x+157$
• $y^2=136x^6+129x^5+117x^4+15x^3+89x^2+18x$
• $y^2=30x^6+79x^5+63x^4+60x^3+131x^2+37x+22$
• $y^2=45x^6+64x^5+9x^4+43x^3+2x^2+5x+155$
• $y^2=120x^6+108x^5+43x^4+49x^3+52x^2+141x+157$
• $y^2=162x^6+130x^5+40x^4+76x^3+49x^2+50x+7$
• $y^2=106x^6+34x^5+94x^4+126x^3+112x^2+23x+35$
• $y^2=16x^6+82x^5+148x^4+119x^3+160x^2+10x+145$
• $y^2=168x^6+110x^5+95x^4+64x^3+159x^2+145x+123$
• $y^2=71x^6+147x^5+24x^4+36x^3+139x^2+102x+33$
• $y^2=171x^6+45x^5+96x^4+149x^3+120x^2+45x+126$
• $y^2=25x^6+101x^5+76x^4+25x^3+90x^2+3x+19$
• $y^2=127x^6+95x^5+30x^4+143x^3+108x^2+48x+78$
• $y^2=115x^6+95x^5+6x^4+10x^3+113x^2+121x+101$
• $y^2=42x^6+137x^5+98x^4+39x^3+99x^2+40x+4$
• $y^2=103x^6+151x^5+99x^4+148x^3+61x^2+20x+163$
• $y^2=75x^6+71x^5+116x^4+55x^3+135x^2+22x+53$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{173^{6}}$.

Endomorphism algebra over $\F_{173}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).

Endomorphism algebra over $\overline{\F}_{173}$

The base change of $A$ to $\F_{173^{6}}$ is 1.26808753332089.nhlic^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{173^{2}}$

  The base change of $A$ to $\F_{173^{2}}$ is the simple isogeny class 2.29929.amr_elwa and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).

• Endomorphism algebra over $\F_{173^{3}}$

  The base change of $A$ to $\F_{173^{3}}$ is the simple isogeny class 2.5177717.a_nhlic and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).

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
2.173.bt_bgq	$2$	(not in LMFDB)
2.173.a_mr	$3$	(not in LMFDB)
2.173.bt_bgq	$3$	(not in LMFDB)