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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-83200.3-e
Number of curves 8
Graph
Conductor 83200.3
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([0,0]),K([0,-1]),K([0,0]),K([3528,2584]),K([-89568,63744])]) 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 83200.3-e 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 83200.3-e over \(\Q(\sqrt{-1}) \)

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

Isogeny class 83200.3-e contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
83200.3-e1 \( \bigl[0\) , \( -i\) , \( 0\) , \( 2584 i + 3528\) , \( 63744 i - 89568\bigr] \)
83200.3-e2 \( \bigl[0\) , \( -i\) , \( 0\) , \( 24 i + 8\) , \( 256 i - 224\bigr] \)
83200.3-e3 \( \bigl[0\) , \( -i\) , \( 0\) , \( -7576 i - 2312\) , \( 242240 i - 526880\bigr] \)
83200.3-e4 \( \bigl[0\) , \( i\) , \( 0\) , \( -1736 i - 152\) , \( 17536 i - 10912\bigr] \)
83200.3-e5 \( \bigl[0\) , \( i\) , \( 0\) , \( 2744 i + 3528\) , \( -66432 i + 81376\bigr] \)
83200.3-e6 \( \bigl[0\) , \( i\) , \( 0\) , \( 15624 i + 9368\) , \( -62656 i - 885856\bigr] \)
83200.3-e7 \( \bigl[0\) , \( i\) , \( 0\) , \( -616 i + 8\) , \( -3840 i + 4320\bigr] \)
83200.3-e8 \( \bigl[0\) , \( -i\) , \( 0\) , \( -9736 i + 168\) , \( 254592 i - 271456\bigr] \)