# Properties

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

# Learn more about

## Invariants

 Base field: $\F_{199}$ Dimension: $2$ L-polynomial: $1 - 49 x + 985 x^{2} - 9751 x^{3} + 39601 x^{4}$ Frobenius angles: $\pm0.0229683311136$, $\pm0.235127646112$ Angle rank: $2$ (numerical) Number field: 4.0.4193837.1 Galois group: $D_{4}$ Jacobians: 7

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 7 curves, and hence is principally polarizable:

• $y^2=156x^6+169x^5+12x^4+110x^3+15x^2+73x+68$
• $y^2=189x^6+192x^5+109x^4+80x^3+102x^2+38x+140$
• $y^2=84x^6+137x^5+87x^4+172x^3+182x^2+81x+75$
• $y^2=173x^6+90x^5+182x^4+171x^3+13x^2+97x+195$
• $y^2=45x^6+74x^5+165x^4+161x^3+95x^2+19x+150$
• $y^2=101x^6+36x^5+112x^4+4x^3+167x^2+193x+10$
• $y^2=68x^6+146x^5+95x^4+20x^3+101x^2+105x+162$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30787 1551264569 62087233421233 2459377163730601517 97393648060043324074672 3856886399424933154053317489 152736579144543489017842677797341 6048521400432541495428011514379268693 239527496481410100073852232030117525066551 9485528389571355695607293529241034752885603584

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 151 39171 7878493 1568241099 312079507116 62103830652615 12358663986210931 2459374186004953635 489415464042499331029 97393677358945312713886

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The endomorphism algebra of this simple isogeny class is 4.0.4193837.1.
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.bx_blx $2$ (not in LMFDB)