# Properties

 Label 2.199.aby_bnj Base Field $\F_{199}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{199}$ Dimension: $2$ L-polynomial: $( 1 - 25 x + 199 x^{2} )^{2}$ Frobenius angles: $\pm0.153403448314$, $\pm0.153403448314$ Angle rank: $1$ (numerical) Jacobians: 47

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 47 curves, and hence is principally polarizable:

• $y^2=97x^6+180x^5+130x^4+148x^3+130x^2+180x+97$
• $y^2=153x^6+66x^5+x^4+187x^3+x^2+66x+153$
• $y^2=2x^6+186x^5+126x^4+40x^3+126x^2+186x+2$
• $y^2=38x^6+9x^5+187x^4+78x^3+162x^2+160x+184$
• $y^2=190x^6+57x^5+166x^4+193x^3+166x^2+57x+190$
• $y^2=3x^6+55x^3+156$
• $y^2=32x^6+146x^5+102x^4+18x^3+148x^2+2x+121$
• $y^2=37x^6+197x^5+68x^4+195x^3+65x^2+157x+104$
• $y^2=142x^6+50x^5+16x^4+101x^3+16x^2+50x+142$
• $y^2=147x^6+188x^5+66x^4+66x^3+66x^2+188x+147$
• $y^2=135x^6+51x^5+90x^4+120x^3+116x^2+99x+30$
• $y^2=27x^6+118x^5+19x^4+130x^3+110x^2+3x+75$
• $y^2=138x^6+97x^5+195x^4+43x^3+195x^2+97x+138$
• $y^2=162x^6+11x^5+65x^3+11x+162$
• $y^2=3x^6+3x^3+176$
• $y^2=115x^6+117x^5+179x^4+186x^3+179x^2+117x+115$
• $y^2=48x^6+107x^5+34x^4+79x^3+34x^2+107x+48$
• $y^2=120x^6+128x^5+150x^4+162x^3+193x^2+177x+73$
• $y^2=135x^6+88x^5+158x^4+22x^3+69x^2+40x+19$
• $y^2=173x^6+175x^5+91x^4+8x^3+34x^2+22x+39$
• and 27 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30625 1550390625 62092824010000 2459460991222265625 97394196115326732015625 3856888913921733456900000000 152736588113505724714099978875625 6048521425738350231493433654253515625 239527496534748430982139499548146308090000 9485528389631399867434823365104451554931640625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 150 39148 7879200 1568294548 312081263250 62103871141198 12358664711933550 2459374196294485348 489415464151483077600 97393677359561822670748

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The isogeny class factors as 1.199.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-19})$$$)$
All geometric endomorphisms are defined over $\F_{199}$.

## 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 2.199.a_ait $2$ (not in LMFDB) 2.199.by_bnj $2$ (not in LMFDB) 2.199.z_qk $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.199.a_ait $2$ (not in LMFDB) 2.199.by_bnj $2$ (not in LMFDB) 2.199.z_qk $3$ (not in LMFDB) 2.199.a_it $4$ (not in LMFDB) 2.199.az_qk $6$ (not in LMFDB)