Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-42250.9-h
Number of curves 4
Graph
Conductor 42250.9
Rank \( 1 \)

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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,-1]),K([1,1]),K([0,-5]),K([-2,1])]) 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 42250.9-h have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 7 & 21 & 3 \\ 7 & 1 & 3 & 21 \\ 21 & 3 & 1 & 7 \\ 3 & 21 & 7 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 42250.9-h over \(\Q(\sqrt{-1}) \)

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

Isogeny class 42250.9-h contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
42250.9-h1 \( \bigl[i\) , \( -i - 1\) , \( i + 1\) , \( -5 i\) , \( i - 2\bigr] \)
42250.9-h2 \( \bigl[1\) , \( i + 1\) , \( i + 1\) , \( 45 i + 399\) , \( -2754 i + 866\bigr] \)
42250.9-h3 \( \bigl[1\) , \( i + 1\) , \( i + 1\) , \( 3285 i - 2281\) , \( -93162 i + 6122\bigr] \)
42250.9-h4 \( \bigl[i\) , \( -i - 1\) , \( i + 1\) , \( -290 i - 5\) , \( 1508 i - 1401\bigr] \)