# Properties

 Label 2.19.al_cm 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: $1 - 11 x + 64 x^{2} - 209 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.165808925970$, $\pm0.370946582130$ Angle rank: $2$ (numerical) Number field: 4.0.349112.1 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:

• $y^2=14x^6+17x^5+8x^4+9x^3+9x^2+16x+14$
• $y^2=18x^6+10x^5+10x^4+x^3+15x^2+15x+2$
• $y^2=3x^6+17x^5+7x^4+9x^3+11x^2+11x+18$
• $y^2=x^6+2x^5+5x^4+9x^3+10x^2+11x+14$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 206 133076 48115832 17034792608 6130763962866 2213470309936064 799069449922614818 288448112752124296832 104127537296738062291928 37589940421461346132563156

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 369 7014 130713 2475979 47049186 893941953 16983957489 322688277282 6131060869489

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.349112.1.
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.l_cm $2$ (not in LMFDB)