# Properties

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

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 2 curves, and hence is principally polarizable:

• $y^2=10x^6+7x^5+6x^4+13x^3+6x^2+7x+10$
• $y^2=2x^6+2x^3+14$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 169 123201 47831056 17140831929 6146645853049 2214310804183296 799032354409893409 288437997911468525289 104126719963943210135056 37589913051239921148065601

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 6 340 6972 131524 2482386 47067046 893900454 16983361924 322685744388 6131056405300

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ah 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
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_al $2$ (not in LMFDB) 2.19.o_dj $2$ (not in LMFDB) 2.19.ai_bt $3$ (not in LMFDB) 2.19.ac_bn $3$ (not in LMFDB) 2.19.b_as $3$ (not in LMFDB) 2.19.h_be $3$ (not in LMFDB) 2.19.q_dy $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.a_al $2$ (not in LMFDB) 2.19.o_dj $2$ (not in LMFDB) 2.19.ai_bt $3$ (not in LMFDB) 2.19.ac_bn $3$ (not in LMFDB) 2.19.b_as $3$ (not in LMFDB) 2.19.h_be $3$ (not in LMFDB) 2.19.q_dy $3$ (not in LMFDB) 2.19.a_l $4$ (not in LMFDB) 2.19.aq_dy $6$ (not in LMFDB) 2.19.ap_dq $6$ (not in LMFDB) 2.19.aj_bu $6$ (not in LMFDB) 2.19.ah_be $6$ (not in LMFDB) 2.19.ag_bf $6$ (not in LMFDB) 2.19.ab_as $6$ (not in LMFDB) 2.19.a_aba $6$ (not in LMFDB) 2.19.a_bl $6$ (not in LMFDB) 2.19.c_bn $6$ (not in LMFDB) 2.19.g_bf $6$ (not in LMFDB) 2.19.i_bt $6$ (not in LMFDB) 2.19.j_bu $6$ (not in LMFDB) 2.19.p_dq $6$ (not in LMFDB) 2.19.a_abl $12$ (not in LMFDB) 2.19.a_ba $12$ (not in LMFDB)