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Properties

Base field \(\Q(\sqrt{85}) \)
Label 2.2.85.1-25.1-b
Number of curves 4
Graph
Conductor 25.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{85}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-21, -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 \( x^{2} - x - 21 \); class number \(2\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,1]),K([1,-1]),K([0,0]),K([-16,-2]),K([6,1])]) 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 25.1-b have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 2 & 6 & 3 \\ 2 & 1 & 3 & 6 \\ 6 & 3 & 1 & 2 \\ 3 & 6 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 25.1-b over \(\Q(\sqrt{85}) \)

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

Isogeny class 25.1-b contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
25.1-b1 \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -2 a - 16\) , \( a + 6\bigr] \)
25.1-b2 \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -2 a + 9\) , \( -4 a - 4\bigr] \)
25.1-b3 \( \bigl[a\) , \( 0\) , \( a\) , \( -987 a - 4059\) , \( 34022 a + 139821\bigr] \)
25.1-b4 \( \bigl[a\) , \( 0\) , \( a\) , \( -15787 a - 64884\) , \( 2283957 a + 9386541\bigr] \)