Properties

Base field \(\Q(\sqrt{-111}) \)
Label 2.0.111.1-196.5-f
Number of curves 6
Graph
Conductor 196.5
Rank \( 1 \)

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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([1,0]),K([-1,0]),K([1,0]),K([-1535,0]),K([23591,0])]) 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 196.5-f have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrr} 1 & 9 & 3 & 6 & 18 & 2 \\ 9 & 1 & 3 & 6 & 2 & 18 \\ 3 & 3 & 1 & 2 & 6 & 6 \\ 6 & 6 & 2 & 1 & 3 & 3 \\ 18 & 2 & 6 & 3 & 1 & 9 \\ 2 & 18 & 6 & 3 & 9 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 196.5-f over \(\Q(\sqrt{-111}) \)

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

Isogeny class 196.5-f contains 6 curves linked by isogenies of degrees dividing 18.

Curve label Weierstrass Coefficients
196.5-f1 \( \bigl[1\) , \( -1\) , \( 1\) , \( -1535\) , \( 23591\bigr] \)
196.5-f2 \( \bigl[1\) , \( -1\) , \( 1\) , \( -5\) , \( -7\bigr] \)
196.5-f3 \( \bigl[1\) , \( -1\) , \( 1\) , \( 40\) , \( 155\bigr] \)
196.5-f4 \( \bigl[1\) , \( -1\) , \( 1\) , \( -320\) , \( 1883\bigr] \)
196.5-f5 \( \bigl[1\) , \( -1\) , \( 1\) , \( -95\) , \( -331\bigr] \)
196.5-f6 \( \bigl[1\) , \( -1\) , \( 1\) , \( -24575\) , \( 1488935\bigr] \)