Elliptic curves - Revision history

A WASSERMANN at 10:10, 8 July 2026

2026-07-08T10:10:07Z

← Older revision		Revision as of 10:10, 8 July 2026
Line 7:		Line 7:
	(function() {		(function() {
	const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
	var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});		var b = board.create('slider',[[1,-3], [10,-3], [-10,2.10,10]], {name:'a'});
	var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var a = board.create('slider',[[1,-4], [10,-4], [-10,-9.52,10]], {name:'b'});

	var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],		var c = board.create('implicitcurve', [(x, y) => x * x * x + a.Value() * x + b.Value() - y * y],
	{strokeWidth:3, strokeColor:'black'});		{strokeWidth:3, strokeColor:'black'});
	})();		})();
Line 18:		Line 18:
	<source lang="javascript">		<source lang="javascript">
	const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
	var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});		var b = board.create('slider',[[1,-3], [10,-3], [-10,2.10,10]], {name:'a'});
	var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var a = board.create('slider',[[1,-4], [10,-4], [-10,-9.52,10]], {name:'b'});

	var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],		var c = board.create('implicitcurve', [(x, y) => xxx + a.Value() * x + b.Value() - y * y],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	</source>		</source>
Line 35:		Line 35:
	(function() {		(function() {
	const board = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});		const board = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});
	var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});		var b = board.create('slider',[[1,-3], [10,-3], [-10,2.10,10]],{name:'a'});
	var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var a = board.create('slider',[[1,-4], [10,-4], [-10,-9.52,10]],{name:'b'});

	var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],
Line 47:		Line 47:
	<source lang="javascript">		<source lang="javascript">
	const board = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});		const board = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});
	var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});		var b = board.create('slider',[[1,-3], [10,-3], [-10,2.10,10]], {name:'a'});
	var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var a = board.create('slider',[[1,-4], [10,-4], [-10,-9.52,10]], {name:'b'});

	var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],

A WASSERMANN at 10:08, 8 July 2026

2026-07-08T10:08:30Z

← Older revision		Revision as of 10:08, 8 July 2026
Line 5:		Line 5:

	<jsxgraph width="600" height="400" box="jxgbox">		<jsxgraph width="600" height="400" box="jxgbox">
	~~JXG.Options.axis.strokeColor = '#cccccc';~~
	(function() {		(function() {
	const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
Line 12:		Line 11:

	var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],		var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3, strokeColor:'black'});
	})();		})();
	</jsxgraph>		</jsxgraph>


			<source lang="javascript">
			const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
			var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
			var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

			var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],
			{strokeWidth:3,strokeColor:'black'});
			</source>

	== Using two function graphs ==		== Using two function graphs ==
Line 30:		Line 39:

	var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3, strokeColor:'black'});
	var c2 = board.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c2 = board.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3, strokeColor:'black'});
	})();		})();
	</jsxgraph>		</jsxgraph>

	<source lang="javascript">		<source lang="javascript">
			const board = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});
			var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
			var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

	~~brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});~~		var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	~~var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});~~		{strokeWidth:3, strokeColor:'black'});
	~~var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});~~		var c2 = board.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],
			{strokeWidth:3, strokeColor:'black'});
	var c1 = ~~brd~~.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});
	var c2 = ~~brd~~.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});
	</source>		</source>

A WASSERMANN at 10:06, 8 July 2026

2026-07-08T10:06:39Z

← Older revision		Revision as of 10:06, 8 July 2026
Line 19:		Line 19:

	Use		Use
	:<math> y^2=x^3+ax+b \quad\~~Longrigrtarrow~~\quad y=\pm\sqrt{x^3+ax+b} </math>		:<math> y^2=x^3+ax+b \quad\Longrightarrow\quad y=\pm\sqrt{x^3+ax+b} </math>

	This approach is somewhat outdated.		This approach is somewhat outdated.

	<jsxgraph width="600" height="400" box="jxgbox1">		<jsxgraph width="600" height="400" box="jxgbox1">
	~~brd~~ = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});		(function() {
	~~brd.suspendUpdate();~~		const board = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});
	var b = ~~brd~~.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});		var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
	var a = ~~brd~~.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

	var c1 = ~~brd~~.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c1 = board.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	var c2 = ~~brd~~.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c2 = board.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	~~brd.unsuspendUpdate~~();		})();
	</jsxgraph>		</jsxgraph>

	<source lang="javascript">		<source lang="javascript">
	~~JXG.Options.axis.strokeColor = '#cccccc';~~
	brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
	var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});		var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});

A WASSERMANN at 10:04, 8 July 2026

2026-07-08T10:04:34Z

← Older revision		Revision as of 10:04, 8 July 2026
Line 6:		Line 6:
	<jsxgraph width="600" height="400" box="jxgbox">		<jsxgraph width="600" height="400" box="jxgbox">
	JXG.Options.axis.strokeColor = '#cccccc';		JXG.Options.axis.strokeColor = '#cccccc';
	const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		(function() {
	var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});		const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
	var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
			var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

	var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],		var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
			})();
	</jsxgraph>		</jsxgraph>

A WASSERMANN at 10:03, 8 July 2026

2026-07-08T10:03:42Z

← Older revision		Revision as of 10:03, 8 July 2026
Line 10:		Line 10:
	var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

	var c1 = board.create('implicitcurve', [(x, y) => ~~Math.sqrt(~~xxx+a.Value()*x+b.Value())],		var c = board.create('implicitcurve', [(x, y) => xxx+a.Value()x+b.Value() - y y],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	</jsxgraph>		</jsxgraph>

A WASSERMANN at 10:02, 8 July 2026

2026-07-08T10:02:43Z

← Older revision		Revision as of 10:02, 8 July 2026
Line 1:		Line 1:
	An elliptic curve can be written as a plane algebraic curve defined by an equation of the form		An elliptic curve can be written as a plane algebraic curve defined by an equation of the form
	:<math> y^2=x^3+ax+b\,. </math>		:<math> y^2=x^3+ax+b\,. </math>

			== Using `implicitcurve` ==

	<jsxgraph width="600" height="400" box="jxgbox">		<jsxgraph width="600" height="400" box="jxgbox">
	JXG.Options.axis.strokeColor = '#cccccc';		JXG.Options.axis.strokeColor = '#cccccc';
	brd = JXG.JSXGraph.initBoard('~~jxgbox~~', {boundingbox: [-15, 10, 15, -10], axis:true});		const board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
			var b = board.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
			var a = board.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

			var c1 = board.create('implicitcurve', [(x, y) => Math.sqrt(xxx+a.Value()*x+b.Value())],
			{strokeWidth:3,strokeColor:'black'});
			</jsxgraph>

			== Using two function graphs ==

			Use
			:<math> y^2=x^3+ax+b \quad\Longrigrtarrow\quad y=\pm\sqrt{x^3+ax+b} </math>

			This approach is somewhat outdated.

			<jsxgraph width="600" height="400" box="jxgbox1">
			brd = JXG.JSXGraph.initBoard('jxgbox1', {boundingbox: [-15, 10, 15, -10], axis:true});
	brd.suspendUpdate();		brd.suspendUpdate();
	var b = brd.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});		var b = brd.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
Line 18:		Line 37:
	JXG.Options.axis.strokeColor = '#cccccc';		JXG.Options.axis.strokeColor = '#cccccc';
	brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
	~~brd.suspendUpdate();~~
	var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});		var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});
	var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});		var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});
Line 26:		Line 44:
	var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	~~brd.unsuspendUpdate();~~
	</source>		</source>

A WASSERMANN at 11:42, 28 February 2014

2014-02-28T11:42:52Z

← Older revision		Revision as of 11:42, 28 February 2014
Line 10:		Line 10:
	var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	//var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	// {strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	brd.unsuspendUpdate();		brd.unsuspendUpdate();
	</jsxgraph>		</jsxgraph>

A WASSERMANN at 11:19, 28 February 2014

2014-02-28T11:19:57Z

← Older revision		Revision as of 11:19, 28 February 2014
Line 5:		Line 5:
	brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
	brd.suspendUpdate();		brd.suspendUpdate();
	var b = brd.create('slider',[[1,-3],[10,-3],[-10,-3,10]],{name:'a'});		var b = brd.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
	var a = brd.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});		var a = brd.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

A WASSERMANN at 11:19, 28 February 2014

2014-02-28T11:19:24Z

← Older revision		Revision as of 11:19, 28 February 2014
Line 6:		Line 6:
	brd.suspendUpdate();		brd.suspendUpdate();
	var b = brd.create('slider',[[1,-3],[10,-3],[-10,-3,10]],{name:'a'});		var b = brd.create('slider',[[1,-3],[10,-3],[-10,-3,10]],{name:'a'});
	var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});		var a = brd.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});

	var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],

A WASSERMANN at 11:18, 28 February 2014

2014-02-28T11:18:25Z

← Older revision		Revision as of 11:18, 28 February 2014
Line 5:		Line 5:
	brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});		brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
	brd.suspendUpdate();		brd.suspendUpdate();
	var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});		var b = brd.create('slider',[[1,-3],[10,-3],[-10,-3,10]],{name:'a'});
	var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});		var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});

	var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],		var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		{strokeWidth:3,strokeColor:'black'});
	var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],		//var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(xxx+a.Value()*x+b.Value()); }],
	{strokeWidth:3,strokeColor:'black'});		// {strokeWidth:3,strokeColor:'black'});
	brd.unsuspendUpdate();		brd.unsuspendUpdate();
	</jsxgraph>		</jsxgraph>