Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-8450.1-b
Number of curves 4
Graph
Conductor 8450.1
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([0,1]),K([-1,0]),K([0,0]),K([-362,-1741]),K([15070,-26365])]) 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 8450.1-b have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 2 & 10 & 5 \\ 2 & 1 & 5 & 10 \\ 10 & 5 & 1 & 2 \\ 5 & 10 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 8450.1-b over \(\Q(\sqrt{-1}) \)

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

Isogeny class 8450.1-b contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
8450.1-b1 \( \bigl[i\) , \( -1\) , \( 0\) , \( -1741 i - 362\) , \( -26365 i + 15070\bigr] \)
8450.1-b2 \( \bigl[1\) , \( 1\) , \( 0\) , \( -86 i - 277\) , \( -498 i - 2511\bigr] \)
8450.1-b3 \( \bigl[i\) , \( -i + 1\) , \( i\) , \( 451 i + 27\) , \( -2160 i - 2404\bigr] \)
8450.1-b4 \( \bigl[1\) , \( i - 1\) , \( 1\) , \( 7071 i + 366\) , \( 147448 i + 171820\bigr] \)