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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-52650.3-a
Number of curves 8
Graph
Conductor 52650.3
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([1,0]),K([-1,0]),K([1,1]),K([-1985,-1454]),K([-26892,-37787])]) 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 52650.3-a have rank \( 1 \).

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 52650.3-a over \(\Q(\sqrt{-1}) \)

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

Isogeny class 52650.3-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
52650.3-a1 \( \bigl[1\) , \( -1\) , \( i + 1\) , \( -1454 i - 1985\) , \( -37787 i - 26892\bigr] \)
52650.3-a2 \( \bigl[1\) , \( -1\) , \( i + 1\) , \( -14 i - 5\) , \( -95 i - 108\bigr] \)
52650.3-a3 \( \bigl[1\) , \( -1\) , \( i + 1\) , \( 4261 i + 1300\) , \( -222278 i - 102195\bigr] \)
52650.3-a4 \( \bigl[i\) , \( 1\) , \( i + 1\) , \( 976 i + 86\) , \( -4604 i - 7398\bigr] \)
52650.3-a5 \( \bigl[i\) , \( 1\) , \( i + 1\) , \( -1544 i - 1984\) , \( 34330 i + 28026\bigr] \)
52650.3-a6 \( \bigl[i\) , \( 1\) , \( i + 1\) , \( -8789 i - 5269\) , \( -373721 i + 26433\bigr] \)
52650.3-a7 \( \bigl[i\) , \( 1\) , \( i + 1\) , \( 346 i - 4\) , \( 1822 i + 1620\bigr] \)
52650.3-a8 \( \bigl[1\) , \( -1\) , \( i + 1\) , \( 5476 i - 95\) , \( -114521 i - 107406\bigr] \)