# Properties

 Label 2.199.abx_bmg 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: $1 - 49 x + 994 x^{2} - 9751 x^{3} + 39601 x^{4}$ Frobenius angles: $\pm0.109464524120$, $\pm0.207308250412$ Angle rank: $2$ (numerical) Number field: 4.0.1912313.2 Galois group: $D_{4}$ Jacobians: 24

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 24 curves, and hence is principally polarizable:

• $y^2=91x^6+63x^5+55x^4+92x^3+33x^2+34x+136$
• $y^2=10x^6+41x^5+119x^4+168x^3+72x^2+192x+9$
• $y^2=119x^6+104x^5+191x^4+111x^3+51x^2+146x+173$
• $y^2=114x^6+160x^5+111x^4+2x^3+14x^2+8x+141$
• $y^2=156x^6+46x^5+61x^4+54x^3+94x^2+163x+90$
• $y^2=39x^6+x^5+137x^4+196x^3+27x^2+65x+2$
• $y^2=20x^6+6x^5+141x^4+94x^3+119x^2+175x+54$
• $y^2=140x^6+21x^5+144x^4+36x^3+76x^2+108x+87$
• $y^2=151x^6+85x^5+36x^4+34x^3+174x^2+92x+76$
• $y^2=146x^6+119x^5+68x^4+120x^3+178x^2+92x+146$
• $y^2=75x^6+116x^5+190x^4+108x^3+5x^2+3x+80$
• $y^2=94x^6+149x^5+130x^4+37x^3+180x^2+31x+145$
• $y^2=99x^6+193x^5+105x^4+182x^3+158x^2+55x$
• $y^2=172x^6+81x^5+22x^4+36x^3+98x^2+143x+62$
• $y^2=68x^6+135x^5+93x^4+177x^3+190x^2+46x+137$
• $y^2=126x^6+173x^5+75x^4+169x^3+84x^2+102x+175$
• $y^2=45x^6+81x^5+146x^4+164x^3+153x^2+99x+58$
• $y^2=105x^6+76x^5+30x^4+150x^3+82x^2+138x+134$
• $y^2=x^6+167x^5+157x^4+194x^3+142x^2+135x+73$
• $y^2=29x^6+101x^5+198x^4+149x^3+73x^2+98x+90$
• $y^2=38x^6+47x^5+184x^4+79x^3+50x^2+123x+76$
• $y^2=51x^6+191x^5+164x^4+53x^3+112x^2+152x+165$
• $y^2=49x^6+5x^5+59x^4+191x^3+24x^2+41x+16$
• $y^2=164x^6+174x^5+183x^4+175x^3+56x^2+105x+193$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30796 1551995216 62097671921392 2459456857100257088 97394075393379673594516 3856888185376959360802091264 152736585224623379978132958633988 6048521417527256449456481434553046272 239527496520758674706290324384242643627888 9485528389643302343976287503677148608831602256

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 151 39189 7879816 1568291913 312080876421 62103859410126 12358664478179947 2459374192955793009 489415464122898455128 97393677359684032621789

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The endomorphism algebra of this simple isogeny class is 4.0.1912313.2.
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.bx_bmg $2$ (not in LMFDB)