# Properties

 Label 2.199.aca_bpi 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: $( 1 - 26 x + 199 x^{2} )^{2}$ Frobenius angles: $\pm0.126927281034$, $\pm0.126927281034$ Angle rank: $1$ (numerical) Jacobians: 24

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 24 curves, and hence is principally polarizable:

• $y^2=x^6+102x^3+140$
• $y^2=86x^6+92x^5+113x^4+129x^3+53x^2+197x+24$
• $y^2=96x^6+171x^5+15x^4+50x^3+113x^2+41x+27$
• $y^2=22x^6+109x^5+193x^4+122x^3+193x^2+109x+22$
• $y^2=188x^6+170x^5+72x^4+99x^3+72x^2+170x+188$
• $y^2=137x^6+66x^5+194x^4+31x^3+144x^2+123x+14$
• $y^2=62x^6+112x^4+112x^2+62$
• $y^2=82x^6+x^5+182x^4+16x^3+50x^2+19x+147$
• $y^2=115x^6+2x^5+144x^4+153x^3+162x^2+158x+196$
• $y^2=110x^6+64x^5+65x^4+77x^3+182x^2+35x+33$
• $y^2=190x^6+136x^5+79x^4+117x^3+79x^2+136x+190$
• $y^2=123x^6+86x^5+96x^4+98x^3+75x^2+35x+1$
• $y^2=166x^6+146x^5+109x^4+60x^3+109x^2+146x+166$
• $y^2=115x^6+20x^5+128x^4+97x^3+117x^2+184x+122$
• $y^2=25x^6+45x^5+20x^4+94x^3+10x^2+157x+38$
• $y^2=77x^6+156x^5+17x^4+54x^3+124x^2+158x+179$
• $y^2=69x^6+35x^5+190x^4+92x^3+133x^2+50x+6$
• $y^2=184x^6+36x^5+164x^4+107x^3+67x^2+100x+57$
• $y^2=190x^6+25x^5+140x^4+104x^3+88x^2+164x+56$
• $y^2=154x^6+96x^5+64x^4+81x^3+64x^2+96x+154$
• $y^2=x^6+47x^3+144$
• $y^2=95x^6+132x^5+53x^4+66x^3+125x^2+72x+170$
• $y^2=67x^6+25x^5+150x^4+18x^3+150x^2+25x+67$
• $y^2=44x^6+186x^4+186x^2+44$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30276 1546376976 62071487074116 2459380210458854400 97393963608677793828996 3856888450761623731040695056 152736587926866928489716907930116 6048521429487008086770538652470886400 239527496557935310494990240966801220071876 9485528389724983886821259872290088762594592016

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 148 39046 7876492 1568243038 312080518228 62103863683366 12358664696831692 2459374197818717758 489415464198859756948 97393677360522706600006

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The isogeny class factors as 1.199.aba 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-30})$$$)$
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_aks $2$ (not in LMFDB) 2.199.ca_bpi $2$ (not in LMFDB) 2.199.ba_sj $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.199.a_aks $2$ (not in LMFDB) 2.199.ca_bpi $2$ (not in LMFDB) 2.199.ba_sj $3$ (not in LMFDB) 2.199.a_ks $4$ (not in LMFDB) 2.199.aba_sj $6$ (not in LMFDB)