# Properties

 Label 2.19.al_co 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: $( 1 - 7 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$ Frobenius angles: $\pm0.203259864187$, $\pm0.348268167089$ Angle rank: $2$ (numerical) Jacobians: 6

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 6 curves, and hence is principally polarizable:

• $y^2=13x^6+x^5+5x^4+15x^3+18x^2+7x$
• $y^2=18x^6+9x^5+3x^4+9x^3+8x^2+18x+2$
• $y^2=4x^5+6x^4+10x^3+15x^2+3x+3$
• $y^2=10x^6+4x^5+x^4+3x^3+17x^2+8x+9$
• $y^2=18x^6+8x^5+17x^4+12x^3+18x^2+15x+15$
• $y^2=15x^6+16x^5+7x^4+14x^3+14x^2+9$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 208 134784 48577984 17093306880 6133488510448 2213192742598656 799009155384017968 288443170343851960320 104127369536184992880064 37589944905494015471858304

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 373 7080 131161 2477079 47043286 893874501 16983666481 322687757400 6131061600853

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ah $\times$ 1.19.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ad_k $2$ (not in LMFDB) 2.19.d_k $2$ (not in LMFDB) 2.19.l_co $2$ (not in LMFDB) 2.19.af_bq $3$ (not in LMFDB) 2.19.e_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ad_k $2$ (not in LMFDB) 2.19.d_k $2$ (not in LMFDB) 2.19.l_co $2$ (not in LMFDB) 2.19.af_bq $3$ (not in LMFDB) 2.19.e_g $3$ (not in LMFDB) 2.19.am_cs $6$ (not in LMFDB) 2.19.ae_g $6$ (not in LMFDB) 2.19.ad_bi $6$ (not in LMFDB) 2.19.d_bi $6$ (not in LMFDB) 2.19.f_bq $6$ (not in LMFDB) 2.19.m_cs $6$ (not in LMFDB)