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Properties

Base field \(\Q(\sqrt{-527}) \)
Label 2.0.527.1-48.2-b
Number of curves 4
Graph
Conductor 48.2
Rank bounds 0...1

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Base field \(\Q(\sqrt{-527}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([132, -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 \( x^{2} - x + 132 \); class number \(18\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([256,0]),K([140,-71])]) 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 48.2-b satisfy \(0 \le r \le 1\).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 48.2-b over \(\Q(\sqrt{-527}) \)

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

Isogeny class 48.2-b contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
48.2-b1 \( \bigl[a\) , \( 0\) , \( 0\) , \( 256\) , \( -71 a + 140\bigr] \)
48.2-b2 \( \bigl[a\) , \( 0\) , \( 0\) , \( -5 a + 361\) , \( 25 a - 1310\bigr] \)
48.2-b3 \( \bigl[a\) , \( 0\) , \( 0\) , \( -5 a + 356\) , \( 24 a - 1248\bigr] \)
48.2-b4 \( \bigl[a\) , \( 0\) , \( 0\) , \( 298 a - 6323\) , \( -17662 a + 216030\bigr] \)