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Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-63.1-a
Number of curves 8
Graph
Conductor 63.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{-7}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -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 \( x^{2} - x + 2 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-34,0]),K([-217,0])]) 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 63.1-a have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 8 & 4 & 8 & 16 & 16 & 2 & 4 \\ 8 & 1 & 2 & 4 & 2 & 2 & 4 & 8 \\ 4 & 2 & 1 & 2 & 4 & 4 & 2 & 4 \\ 8 & 4 & 2 & 1 & 8 & 8 & 4 & 8 \\ 16 & 2 & 4 & 8 & 1 & 4 & 8 & 16 \\ 16 & 2 & 4 & 8 & 4 & 1 & 8 & 16 \\ 2 & 4 & 2 & 4 & 8 & 8 & 1 & 2 \\ 4 & 8 & 4 & 8 & 16 & 16 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 63.1-a over \(\Q(\sqrt{-7}) \)

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

Isogeny class 63.1-a contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
63.1-a1 \( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \)
63.1-a2 \( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \)
63.1-a3 \( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \)
63.1-a4 \( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \)
63.1-a5 \( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( a - 1\) , \( -a\bigr] \)
63.1-a6 \( \bigl[1\) , \( a\) , \( a\) , \( -2 a + 1\) , \( 0\bigr] \)
63.1-a7 \( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \)
63.1-a8 \( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \)