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### points on elliptic curve number field

I have the following elliptic curve : y^2=x^3-3267x+45630 and generator P=[-21,324].

I want to find the general formula to add any point (r,s) to the point ((-15/2)+(27/2)*B,0) where B^2=17.

PARI seems to be pretty straightforward in giving answer. And i think the mistake I did in SAGE is that I should define r and s in number field which I'm not sure how to do it.

The Pari code is as follows : gp > elladd(E,[r,s],[(-15/2)+(27/2)B,0])= [(-8r^3 + (108B - 60)r^2 + (1458B^2 - 1620B + 450)r + (8s^2 + (-19683B^3 + 32805B^2 - 18225B + 3375)))/(8r^2 + (-216B + 120)r + (1458B^2 - 1620B + 450)), (-8sr^3 + (4374B^2 - 4860B + 1350)sr + (8s^3 + (-39366B^3 + 65610B^2 - 36450B + 6750)s))/(-8r^3 + (324B - 180)r^2 + (-4374B^2 + 4860B - 1350)r + (19683B^3 - 32805B^2 + 18225B - 3375))]

 2 No.2 Revision kcrisman 11972 ●37 ●124 ●241

### points on elliptic curve number field

I have the following elliptic curve : y^2=x^3-3267x+45630 and generator P=[-21,324].

I want to find the general formula to add any point (r,s) to the point ((-15/2)+(27/2)*B,0) where B^2=17.

PARI seems to be pretty straightforward in giving answer. And i think the mistake I did in SAGE is that I should define r and s in number field which I'm not sure how to do it.

The Pari code is as follows : :

gp > elladd(E,[r,s],[(-15/2)+(27/2)B,0])= [(-8r^3 elladd(E,[r,s],[(-15/2)+(27/2)*B,0])= [(-8*r^3 + (108B (108*B - 60)r^2 60)*r^2 + (1458B^2 (1458*B^2 - 1620B 1620*B + 450)r 450)*r + (8s^2 (8*s^2 + (-19683B^3 (-19683*B^3 + 32805B^2 32805*B^2 - 18225B 18225*B + 3375)))/(8r^2 3375)))/(8*r^2 + (-216B (-216*B + 120)r 120)*r + (1458B^2 (1458*B^2 - 1620B 1620*B + 450)), (-8sr^3 (-8*s*r^3 + (4374B^2 (4374*B^2 - 4860B 4860*B + 1350)sr 1350)*s*r + (8s^3 (8*s^3 + (-39366B^3 (-39366*B^3 + 65610B^2 65610*B^2 - 36450B 36450*B + 6750)s))/(-8r^3 6750)*s))/(-8*r^3 + (324B (324*B - 180)r^2 180)*r^2 + (-4374B^2 (-4374*B^2 + 4860B 4860*B - 1350)r 1350)*r + (19683B^3 (19683*B^3 - 32805B^2 32805*B^2 + 18225B 18225*B - 3375))]3375))]