# Properties

 Label 2.173.abt_bgq Base Field $\F_{173}$ Dimension $2$ Ordinary Yes $p$-rank $2$ 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.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 44 curves, 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 24 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 22948 885976384 26808759403456 802402518753628416 24013957048436101180468 718709580752390504984743936 21510249794143156738178611251892 643780251725664619672837476297237504 19267699140703639474908789608000549086144 576662967582142020174554028775116629789103424

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

## Decomposition and endomorphism algebra

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}($$$\Q(\sqrt{-51})$$$)$
All geometric endomorphisms are defined over $\F_{173^{6}}$.
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)
Below is a list of all 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) 2.173.a_mr $6$ (not in LMFDB) 2.173.a_amr $12$ (not in LMFDB)