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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-650.4-a
Number of curves 8
Graph
Conductor 650.4
Rank \( 0 \)

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Base field \(\Q(\sqrt{-1}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 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 \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,0]),K([1,1]),K([-221,161]),K([996,-1400])]) 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 650.4-a have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 12 & 2 & 4 & 6 & 12 \\ 3 & 1 & 12 & 4 & 6 & 12 & 2 & 4 \\ 4 & 12 & 1 & 12 & 2 & 4 & 6 & 3 \\ 12 & 4 & 12 & 1 & 6 & 3 & 2 & 4 \\ 2 & 6 & 2 & 6 & 1 & 2 & 3 & 6 \\ 4 & 12 & 4 & 3 & 2 & 1 & 6 & 12 \\ 6 & 2 & 6 & 2 & 3 & 6 & 1 & 2 \\ 12 & 4 & 3 & 4 & 6 & 12 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 650.4-a over \(\Q(\sqrt{-1}) \)

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

Isogeny class 650.4-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
650.4-a1 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( 161 i - 221\) , \( -1400 i + 996\bigr] \)
650.4-a2 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( i - 1\) , \( -4 i + 4\bigr] \)
650.4-a3 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -474 i + 144\) , \( -8233 i + 3785\bigr] \)
650.4-a4 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -109 i + 9\) , \( 170 i - 274\bigr] \)
650.4-a5 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( 171 i - 221\) , \( -1272 i + 1038\bigr] \)
650.4-a6 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( 976 i - 586\) , \( 13841 i + 979\bigr] \)
650.4-a7 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -39 i - 1\) , \( -68 i + 60\bigr] \)
650.4-a8 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -609 i - 11\) , \( -4242 i + 3978\bigr] \)