︠3adb0611-b8f0-4346-9b8f-1aa7a3065ed0si︠ %hide html('

Elliptic Curves over the Real Numbers

') @interact def _(a = slider([-10..3],default=-4), b = (4,(-10..10))): if 4*a^3 + 27*b^2 == 0: html('The choice of $a$ = %s and $b$ = %s does not define an elliptic curve because this makes $27a^3 + 4b^2 = 0.$'%(a,b)) else: E = EllipticCurve([a,b]) s = '$E(\mathbb{R}) = \{ (x,y) \in \mathbb{R} \\times \mathbb{R}\ |\ %s \} \cup \{ \mathcal{O} \}$'%(latex(E)) html(''+s+'') show(plot(E,xmax=3),axes_labels=['x','y']) html('Figure 3.2 Play around with different values for a and b and note how the curve changes. You can choose a between -10 and
3, and b between -10 and 10, because this is the range where you can see something happening. Also note that this program
only lets you choose integers for a and b, although real numbers are actually allowed for a curve over the real numbers. This
is because real numbers for a and b would make the output ugly.') ︡d0822450-7875-440e-98d7-3958b9a7f3e4︡{"hide":"input"}︡{"hide":"input"}︡{"html":"

Elliptic Curves over the Real Numbers

"}︡{"interact":{"controls":[{"animate":true,"control_type":"slider","default":6,"display_value":true,"label":"a","vals":["-10","-9","-8","-7","-6","-5","-4","-3","-2","-1","0","1","2","3"],"var":"a","width":null},{"animate":true,"control_type":"slider","default":14,"display_value":true,"label":"b","vals":["-10","-9","-8","-7","-6","-5","-4","-3","-2","-1","0","1","2","3","4","5","6","7","8","9","10"],"var":"b","width":null}],"flicker":false,"id":"40308c16-a2da-463d-aec6-7016ad822db7","layout":[[["a",12,null]],[["b",12,null]],[["",12,null]]],"style":"None"}}︡{"hide":"input"}︡{"html":"Figure 3.2 Play around with different values for a and b and note how the curve changes. You can choose a between -10 and
3, and b between -10 and 10, because this is the range where you can see something happening. Also note that this program
only lets you choose integers for a and b, although real numbers are actually allowed for a curve over the real numbers. This
is because real numbers for a and b would make the output ugly."}︡{"done":true}︡ ︠13068e92-7c20-4bbf-a878-50051edcba8es︠ ︡e709bf95-1172-4717-bc7a-80ce4662cf02︡{"done":true}︡