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Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-171.1-j
Number of curves 3
Graph
Conductor 171.1
Rank \( 1 \)

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Base field \(\Q(\sqrt{57}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -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 \( x^{2} - x - 14 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([-1,-1]),K([1,0]),K([-394663,92321]),K([-127121496,29736602])]) 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 171.1-j have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 171.1-j over \(\Q(\sqrt{57}) \)

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

Isogeny class 171.1-j contains 3 curves linked by isogenies of degrees dividing 9.

Curve label Weierstrass Coefficients
171.1-j1 \( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 92321 a - 394663\) , \( 29736602 a - 127121496\bigr] \)
171.1-j2 \( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 1121 a - 4783\) , \( 43022 a - 183921\bigr] \)
171.1-j3 \( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -79 a + 347\) , \( -28 a + 114\bigr] \)