# Properties

 Label 2.19.ao_di 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: $( 1 - 8 x + 19 x^{2} )( 1 - 6 x + 19 x^{2} )$ Frobenius angles: $\pm0.130073469147$, $\pm0.258380448083$ Angle rank: $2$ (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:

• $y^2=12x^6+5x^5+5x^4+3x^3+5x^2+5x+12$
• $y^2=10x^6+6x^5+6x^3+16x+15$
• $y^2=8x^6+17x^5+7x^4+11x^3+7x^2+17x+8$
• $y^2=13x^6+14x^5+18x^4+18x^3+3x^2+12x+10$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 168 122304 47532744 17083422720 6139358981448 2213711304831936 799013955490370088 288441475423798394880 104127474329448490192104 37589999149034860983559104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 338 6930 131086 2479446 47054306 893879874 16983566686 322688082150 6131070448178

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ai $\times$ 1.19.ag 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.ac_ak $2$ (not in LMFDB) 2.19.c_ak $2$ (not in LMFDB) 2.19.o_di $2$ (not in LMFDB) 2.19.af_bg $3$ (not in LMFDB) 2.19.b_ae $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ac_ak $2$ (not in LMFDB) 2.19.c_ak $2$ (not in LMFDB) 2.19.o_di $2$ (not in LMFDB) 2.19.af_bg $3$ (not in LMFDB) 2.19.b_ae $3$ (not in LMFDB) 2.19.an_dc $6$ (not in LMFDB) 2.19.ah_bs $6$ (not in LMFDB) 2.19.ab_ae $6$ (not in LMFDB) 2.19.f_bg $6$ (not in LMFDB) 2.19.h_bs $6$ (not in LMFDB) 2.19.n_dc $6$ (not in LMFDB)