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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-67600.6-a
Number of curves 8
Graph
Conductor 67600.6
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([1,1]),K([1,1]),K([1,1]),K([12162,7355]),K([-474647,455466])]) 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 67600.6-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 67600.6-a over \(\Q(\sqrt{-1}) \)

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

Isogeny class 67600.6-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
67600.6-a1 \( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( 7355 i + 12162\) , \( 455466 i - 474647\bigr] \)
67600.6-a2 \( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( -5 i + 82\) , \( 1082 i - 1719\bigr] \)
67600.6-a3 \( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( 2535 i - 25618\) , \( 2731422 i - 1988155\bigr] \)
67600.6-a4 \( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( 1714 i - 5398\) , \( 43016 i - 113108\bigr] \)
67600.6-a5 \( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( 7154 i + 12642\) , \( -393176 i + 473532\bigr] \)
67600.6-a6 \( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( 8574 i + 58582\) , \( 5164160 i - 636316\bigr] \)
67600.6-a7 \( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( 794 i - 1838\) , \( -20520 i + 26940\bigr] \)
67600.6-a8 \( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( 12675 i - 28998\) , \( 1245458 i - 1781967\bigr] \)