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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-68450.5-h
Number of curves 8
Graph
Conductor 68450.5
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([0,0]),K([0,0]),K([2805,-920]),K([28239,54952])]) 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 68450.5-h have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 6 & 3 & 12 & 2 & 4 & 12 \\ 4 & 1 & 6 & 12 & 3 & 2 & 4 & 12 \\ 6 & 6 & 1 & 2 & 2 & 3 & 6 & 2 \\ 3 & 12 & 2 & 1 & 4 & 6 & 12 & 4 \\ 12 & 3 & 2 & 4 & 1 & 6 & 12 & 4 \\ 2 & 2 & 3 & 6 & 6 & 1 & 2 & 6 \\ 4 & 4 & 6 & 12 & 12 & 2 & 1 & 3 \\ 12 & 12 & 2 & 4 & 4 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 68450.5-h over \(\Q(\sqrt{-1}) \)

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

Isogeny class 68450.5-h contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
68450.5-h1 \( \bigl[i\) , \( 0\) , \( 0\) , \( -920 i + 2805\) , \( 54952 i + 28239\bigr] \)
68450.5-h2 \( \bigl[i\) , \( 0\) , \( 0\) , \( 920 i + 2805\) , \( -54952 i + 28239\bigr] \)
68450.5-h3 \( \bigl[i\) , \( 0\) , \( 0\) , \( -5255\) , \( 149075\bigr] \)
68450.5-h4 \( \bigl[i\) , \( 0\) , \( 0\) , \( 5030 i - 5095\) , \( 293862 i + 186619\bigr] \)
68450.5-h5 \( \bigl[i\) , \( 0\) , \( 0\) , \( -5030 i - 5095\) , \( -293862 i + 186619\bigr] \)
68450.5-h6 \( \bigl[i\) , \( 0\) , \( 0\) , \( 245\) , \( 975\bigr] \)
68450.5-h7 \( \bigl[i\) , \( 0\) , \( 0\) , \( -75\) , \( 143\bigr] \)
68450.5-h8 \( \bigl[i\) , \( 0\) , \( 0\) , \( -5275\) , \( 147903\bigr] \)