# Properties

 Label 2.163.abu_bgx Base Field $\F_{163}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{163}$ Dimension: $2$ L-polynomial: $( 1 - 23 x + 163 x^{2} )^{2}$ Frobenius angles: $\pm0.143017980409$, $\pm0.143017980409$ Angle rank: $1$ (numerical) Jacobians: 12

This isogeny class is not 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 12 curves, and hence is principally polarizable:

• $y^2=54x^6+65x^5+127x^4+49x^3+73x^2+29x+127$
• $y^2=51x^6+51x^5+135x^4+67x^3+135x^2+51x+51$
• $y^2=116x^6+162x^5+96x^4+74x^3+96x^2+162x+116$
• $y^2=29x^6+23x^5+116x^4+152x^3+61x^2+102x+126$
• $y^2=33x^6+39x^5+74x^4+143x^3+74x^2+39x+33$
• $y^2=x^6+54x^3+6$
• $y^2=151x^6+71x^5+159x^4+28x^3+84x^2+138x+19$
• $y^2=10x^6+106x^5+147x^4+5x^3+109x^2+108x+62$
• $y^2=87x^6+125x^5+78x^4+109x^3+78x^2+125x+87$
• $y^2=141x^6+11x^5+120x^4+49x^3+53x^2+66x+62$
• $y^2=132x^6+133x^5+131x^4+30x^3+82x^2+4x+71$
• $y^2=98x^6+28x^5+119x^4+139x^3+90x^2+2x+9$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 19881 695218689 18747410509584 498328257515063481 13239733616627199559641 351764181157756594946754816 9346015416601942599610691534241 248314266904961255998581494146821609 6597461725597123709983336583287845605136 175287960540486808950011013697965826299530049

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 118 26164 4328908 705935620 115064465698 18755385208198 3057125462379190 498311416857166084 81224760556202090164 13239635967118307726164

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The isogeny class factors as 1.163.ax 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-123})$$$)$
All geometric endomorphisms are defined over $\F_{163}$.

## 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 2.163.a_ahv $2$ (not in LMFDB) 2.163.bu_bgx $2$ (not in LMFDB) 2.163.x_oc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.163.a_ahv $2$ (not in LMFDB) 2.163.bu_bgx $2$ (not in LMFDB) 2.163.x_oc $3$ (not in LMFDB) 2.163.a_hv $4$ (not in LMFDB) 2.163.ax_oc $6$ (not in LMFDB)