# Properties

 Label 2.19.al_cp 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: $1 - 11 x + 67 x^{2} - 209 x^{3} + 361 x^{4}$ Frobenius angles: $\pm0.225619498788$, $\pm0.332359937866$ Angle rank: $2$ (numerical) Number field: 4.0.45725.1 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:

• $y^2=12x^6+11x^5+13x^4+x^3+4x^2+5x+10$
• $y^2=15x^6+11x^5+13x^4+16x^3+12x^2+16x+2$
• $y^2=10x^6+6x^5+4x^4+6x^3+16x^2+16x$
• $y^2=18x^6+2x^5+3x^4+13x^3+14x^2+8x+7$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 209 135641 48809651 17121827789 6134444175024 2212955922576641 798965794289625479 288439911158049053909 104127351848432066042141 37589968388064876490181376

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 375 7113 131379 2477464 47038251 893825991 16983474579 322687702587 6131065430950

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is 4.0.45725.1.
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_cp $2$ (not in LMFDB)