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Properties

Base field \(\Q(\sqrt{-35}) \)
Label 2.0.35.1-35.1-a
Number of curves 3
Graph
Conductor 35.1
Rank \( 2 \)

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Base field \(\Q(\sqrt{-35}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, -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 \( x^{2} - x + 9 \); class number \(2\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([-1,1]),K([1,0]),K([1048,131]),K([-11046,5198])]) 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 35.1-a have rank \( 2 \).

Isogeny matrix

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

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 35.1-a over \(\Q(\sqrt{-35}) \)

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

Isogeny class 35.1-a contains 3 curves linked by isogenies of degrees dividing 9.

Curve label Weierstrass Coefficients
35.1-a1 \( \bigl[0\) , \( a - 1\) , \( 1\) , \( 131 a + 1048\) , \( 5198 a - 11046\bigr] \)
35.1-a2 \( \bigl[0\) , \( -a\) , \( 1\) , \( -a + 9\) , \( 2 a + 2\bigr] \)
35.1-a3 \( \bigl[0\) , \( a - 1\) , \( 1\) , \( -9 a - 72\) , \( -10 a + 21\bigr] \)