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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-42250.7-e
Number of curves 8
Graph
Conductor 42250.7
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([1,0]),K([1,-1]),K([0,1]),K([1767,-17680]),K([728133,-549379])]) 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 42250.7-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 42250.7-e over \(\Q(\sqrt{-1}) \)

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

Isogeny class 42250.7-e contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
42250.7-e1 \( \bigl[1\) , \( -i + 1\) , \( i\) , \( -17680 i + 1767\) , \( -549379 i + 728133\bigr] \)
42250.7-e2 \( \bigl[1\) , \( -i + 1\) , \( i\) , \( -80 i + 67\) , \( -2199 i + 1873\bigr] \)
42250.7-e3 \( \bigl[1\) , \( -i + 1\) , \( i\) , \( 23715 i - 21748\) , \( -2017251 i + 4265587\bigr] \)
42250.7-e4 \( \bigl[i\) , \( i - 1\) , \( 1\) , \( 4110 i - 5763\) , \( -148881 i + 93187\bigr] \)
42250.7-e5 \( \bigl[i\) , \( i - 1\) , \( 1\) , \( -18010 i + 2327\) , \( 570821 i - 661327\bigr] \)
42250.7-e6 \( \bigl[i\) , \( i - 1\) , \( 1\) , \( -65015 i + 35362\) , \( 481119 i + 7224437\bigr] \)
42250.7-e7 \( \bigl[i\) , \( i - 1\) , \( 1\) , \( 1240 i - 2173\) , \( 30671 i - 32777\bigr] \)
42250.7-e8 \( \bigl[1\) , \( -i + 1\) , \( i\) , \( 19490 i - 34423\) , \( -2094671 i + 2160277\bigr] \)