# Properties

 Label 2.19.am_cq Base Field $\F_{19}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19}$ Dimension: $2$ L-polynomial: $1 - 12 x + 68 x^{2} - 228 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.0791769653548$, $\pm0.366480294854$ Angle rank: $2$ (numerical) Number field: 4.0.168192.4 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

• $y^2=12x^6+5x^5+3x^4+8x^3+6x^2+13x+7$
• $y^2=18x^6+4x^5+x^4+8x^3+9x^2+10x+11$
• $y^2=4x^6+18x^5+4x^4+13x^3+4x^2+2x+5$
• $y^2=2x^6+14x^5+13x^4+11x^3+3x+8$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 190 127300 47287390 16942611600 6121833155950 2212741065343300 799026649637951470 288447540686741606400 104127794396037832168030 37589982524610807362282500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 354 6896 130006 2472368 47033682 893894072 16983923806 322689074024 6131067736674

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.168192.4.
All geometric endomorphisms are defined over $\F_{19}$.

## 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 2.19.m_cq $2$ (not in LMFDB)