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Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-128.5-j
Number of curves 4
Graph
Conductor 128.5
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([0,1]),K([-1,0]),K([0,0]),K([-82866156,-25303280]),K([-440940033556,-134641581530])]) 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 128.5-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 128.5-j over \(\Q(\sqrt{57}) \)

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

Isogeny class 128.5-j contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
128.5-j1 \( \bigl[a\) , \( -1\) , \( 0\) , \( -25303280 a - 82866156\) , \( -134641581530 a - 440940033556\bigr] \)
128.5-j2 \( \bigl[a\) , \( a\) , \( 0\) , \( 1044512 a - 4465409\) , \( -1129233268 a + 4827376713\bigr] \)
128.5-j3 \( \bigl[a\) , \( -1\) , \( 0\) , \( -9410 a - 30816\) , \( -1409836 a - 4617096\bigr] \)
128.5-j4 \( \bigl[a\) , \( a\) , \( 0\) , \( 392 a - 1649\) , \( -11804 a + 50489\bigr] \)