Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-1980.2-e
Number of curves 4
Graph
Conductor 1980.2
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([0,1]),K([-1,0]),K([1,0]),K([-73,74]),K([601,103])]) 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 1980.2-e have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 1980.2-e over \(\Q(\sqrt{5}) \)

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

Isogeny class 1980.2-e contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
1980.2-e1 \( \bigl[\phi\) , \( -1\) , \( 1\) , \( 74 \phi - 73\) , \( 103 \phi + 601\bigr] \)
1980.2-e2 \( \bigl[1\) , \( \phi + 1\) , \( 0\) , \( 1\) , \( -4 \phi + 5\bigr] \)
1980.2-e3 \( \bigl[1\) , \( \phi + 1\) , \( 0\) , \( 20 \phi - 39\) , \( -56 \phi + 93\bigr] \)
1980.2-e4 \( \bigl[1\) , \( \phi + 1\) , \( 0\) , \( -10 \phi - 129\) , \( 406 \phi - 105\bigr] \)