# Properties

 Label 2.199.aby_bne 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: $1 - 50 x + 1018 x^{2} - 9950 x^{3} + 39601 x^{4}$ Frobenius angles: $\pm0.0840302640260$, $\pm0.201172616085$ Angle rank: $2$ (numerical) Number field: 4.0.376400.1 Galois group: $D_{4}$ Jacobians: 54

This isogeny class is simple and 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 54 curves, and hence is principally polarizable:

• $y^2=169x^6+155x^5+183x^4+105x^3+29x^2+66x$
• $y^2=44x^6+171x^5+62x^4+102x^3+86x^2+71x+15$
• $y^2=46x^6+59x^5+137x^4+197x^3+12x^2+54x+109$
• $y^2=37x^6+115x^5+50x^4+166x^3+157x^2+174x+27$
• $y^2=163x^6+68x^5+39x^4+9x^3+64x^2+167x+164$
• $y^2=27x^6+112x^5+33x^4+x^3+192x^2+124x+196$
• $y^2=70x^6+57x^5+166x^4+33x^3+19x^2+110x+71$
• $y^2=176x^6+191x^5+149x^4+105x^3+140x^2+34x+170$
• $y^2=137x^6+156x^5+43x^4+121x^3+107x^2+59x+41$
• $y^2=19x^6+84x^5+x^4+109x^3+18x^2+69x+31$
• $y^2=33x^6+80x^5+41x^4+102x^3+180x^2+29x+92$
• $y^2=177x^6+16x^5+129x^4+167x^3+141x^2+71x+133$
• $y^2=74x^6+112x^5+132x^4+51x^3+35x^2+79x+21$
• $y^2=77x^6+56x^5+83x^4+66x^3+51x^2+60x+77$
• $y^2=177x^6+36x^5+198x^4+171x^3+75x^2+41x+9$
• $y^2=48x^6+123x^5+139x^4+179x^3+14x^2+119x+17$
• $y^2=176x^6+77x^5+43x^4+48x^3+145x^2+11x+57$
• $y^2=92x^6+53x^5+49x^4+148x^3+21x^2+75x+180$
• $y^2=43x^6+183x^5+170x^4+38x^3+63x^2+126x+110$
• $y^2=168x^6+178x^5+107x^4+85x^3+178x^2+38x+24$
• and 34 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30620 1549984400 62086905995180 2459414583733126400 97393939428854323975500 3856887808961276372867632400 152736584298256252785492947433980 6048521415452027726835637992024166400 239527496516564876640012642344705882740220 9485528389636924418573186328242558172350250000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 150 39138 7878450 1568264958 312080440750 62103853349058 12358664403223050 2459374192111989438 489415464114329461350 97393677359618546590498

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The endomorphism algebra of this simple isogeny class is 4.0.376400.1.
All geometric endomorphisms are defined over $\F_{199}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.199.by_bne $2$ (not in LMFDB)