# Properties

 Label 2.19.am_cw 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: $( 1 - 6 x + 19 x^{2} )^{2}$ Frobenius angles: $\pm0.258380448083$, $\pm0.258380448083$ Angle rank: $1$ (numerical) Jacobians: 4

This isogeny class is not 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:

• $y^2=15x^6+9x^4+9x^2+15$
• $y^2=10x^6+14x^5+9x^4+11x^3+9x^2+14x+10$
• $y^2=18x^6+18x^4+x^3+x^2+3x$
• $y^2=15x^6+5x^5+8x^4+5x^3+14x^2+7x+13$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 196 132496 48804196 17171481600 6140553384196 2213111967788176 798918529220525476 288432756057522585600 104126968026280916546116 37589989261020791711932816

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 366 7112 131758 2479928 47041566 893773112 16983053278 322686513128 6131068835406

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-10})$$$)$
All geometric endomorphisms are defined over $\F_{19}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.a_c $2$ (not in LMFDB) 2.19.m_cw $2$ (not in LMFDB) 2.19.g_r $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.a_c $2$ (not in LMFDB) 2.19.m_cw $2$ (not in LMFDB) 2.19.g_r $3$ (not in LMFDB) 2.19.a_ac $4$ (not in LMFDB) 2.19.ag_r $6$ (not in LMFDB)