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Properties

Base field \(\Q(\sqrt{-111}) \)
Label 2.0.111.1-441.3-d
Number of curves 2
Graph
Conductor 441.3
Rank bounds 0...1

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Base field \(\Q(\sqrt{-111}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([28, -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 \(a\), with minimal polynomial \( x^{2} - x + 28 \); class number \(8\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([-1,-1]),K([1,1]),K([-36,0]),K([51,12])]) E.isogeny_class()
 

Rank

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

The rank \(r\) of the elliptic curves in class 441.3-d satisfy \(0 \le r \le 1\).

Isogeny matrix

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 441.3-d over \(\Q(\sqrt{-111}) \)

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

Isogeny class 441.3-d contains 2 curves linked by isogenies of degree 5.

Curve label Weierstrass Coefficients
441.3-d1 \( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( -36\) , \( 12 a + 51\bigr] \)
441.3-d2 \( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( -210 a - 5706\) , \( -7254 a - 163203\bigr] \)