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Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-722.1-b
Number of curves 3
Graph
Conductor 722.1
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))
 
Copy content pari:K = nfinit(Polrev(%s));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R!%s);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx(%s))
 

Generator \(a\), with minimal polynomial \( x^{2} - 2 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-16,0]),K([22,0])]) E.isogeny_class()
 

Rank

Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 

The elliptic curves in class 722.1-b have rank \( 1 \).

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_class().graph().plot(edge_labels=True)
 

Elliptic curves in class 722.1-b over \(\Q(\sqrt{2}) \)

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 

Isogeny class 722.1-b contains 3 curves linked by isogenies of degrees dividing 9.

Curve label Weierstrass Coefficients
722.1-b1 \( \bigl[1\) , \( 0\) , \( 1\) , \( -16\) , \( 22\bigr] \)
722.1-b2 \( \bigl[1\) , \( 0\) , \( 1\) , \( -86\) , \( -2456\bigr] \)
722.1-b3 \( \bigl[1\) , \( 0\) , \( 1\) , \( 9\) , \( 90\bigr] \)