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Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-128.6-j
Number of curves 4
Graph
Conductor 128.6
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([1,1]),K([1,1]),K([0,0]),K([-3420890,-1044502]),K([3683520340,1124767860])]) 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 128.6-j have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 128.6-j over \(\Q(\sqrt{57}) \)

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

Isogeny class 128.6-j contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
128.6-j1 \( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -1044502 a - 3420890\) , \( 1124767860 a + 3683520340\bigr] \)
128.6-j2 \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 25303280 a - 108169436\) , \( 134641581530 a - 575581615086\bigr] \)
128.6-j3 \( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -382 a - 1250\) , \( 10156 a + 33260\bigr] \)
128.6-j4 \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 9410 a - 40226\) , \( 1409836 a - 6026932\bigr] \)