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Properties

Base field \(\Q(\sqrt{-5}) \)
Label 2.0.20.1-69.3-b
Number of curves 1
Graph
Conductor 69.3
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([5, 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 \( x^{2} + 5 \); class number \(2\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,0]),K([0,1]),K([0,0]),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 69.3-b1 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 69.3-b over \(\Q(\sqrt{-5}) \)

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

Isogeny class 69.3-b contains only one elliptic curve.

Curve label Weierstrass Coefficients
69.3-b1 \( \bigl[1\) , \( 0\) , \( a\) , \( 0\) , \( 1\bigr] \)