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Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-3131.4-a
Number of curves 1
Graph
Conductor 3131.4
Rank \( 1 \)

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Base field \(\Q(\sqrt{5}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -1, 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 \(\phi\), with minimal polynomial \( x^{2} - x - 1 \); class number \(1\).

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

Rank

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

The elliptic curve 3131.4-a1 has rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{r} 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 3131.4-a over \(\Q(\sqrt{5}) \)

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

Isogeny class 3131.4-a contains only one elliptic curve.

Curve label Weierstrass Coefficients
3131.4-a1 \( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( -\phi\) , \( -1\bigr] \)