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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-650.3-a
Number of curves 8
Graph
Conductor 650.3
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,-162]),K([996,1399])]) 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 650.3-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 650.3-a over \(\Q(\sqrt{-1}) \)

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

Isogeny class 650.3-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
650.3-a1 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -162 i - 221\) , \( 1399 i + 996\bigr] \)
650.3-a2 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -2 i - 1\) , \( 3 i + 4\bigr] \)
650.3-a3 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( 473 i + 144\) , \( 8232 i + 3785\bigr] \)
650.3-a4 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( 108 i + 9\) , \( -171 i - 274\bigr] \)
650.3-a5 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -172 i - 221\) , \( 1271 i + 1038\bigr] \)
650.3-a6 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( -977 i - 586\) , \( -13842 i + 979\bigr] \)
650.3-a7 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( 38 i - 1\) , \( 67 i + 60\bigr] \)
650.3-a8 \( \bigl[1\) , \( 0\) , \( i + 1\) , \( 608 i - 11\) , \( 4241 i + 3978\bigr] \)