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Properties

Base field \(\Q(\sqrt{85}) \)
Label 2.2.85.1-25.1-a
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([-85905,-20905]),K([12005575,2921225])]) 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-a 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-a 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-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
25.1-a1 \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -20905 a - 85905\) , \( 2921225 a + 12005575\bigr] \)
25.1-a2 \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -6105 a - 25080\) , \( -521090 a - 2141555\bigr] \)
25.1-a3 \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -1240 a - 5100\) , \( 45281 a + 186092\bigr] \)
25.1-a4 \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -19915 a - 81850\) , \( 3190111 a + 13110627\bigr] \)