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Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-42250.9-i
Number of curves 8
Graph
Conductor 42250.9
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([0,1]),K([1,0]),K([16476,-6646]),K([499366,769829])]) 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 42250.9-i 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 42250.9-i over \(\Q(\sqrt{-1}) \)

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

Isogeny class 42250.9-i contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
42250.9-i1 \( \bigl[1\) , \( i\) , \( 1\) , \( -6646 i + 16476\) , \( 769829 i + 499366\bigr] \)
42250.9-i2 \( \bigl[1\) , \( i\) , \( 1\) , \( -86 i + 56\) , \( 2677 i + 1030\bigr] \)
42250.9-i3 \( \bigl[1\) , \( i\) , \( 1\) , \( 27519 i - 16679\) , \( 3402809 i + 3267006\bigr] \)
42250.9-i4 \( \bigl[i\) , \( -i\) , \( i\) , \( 6684 i - 2333\) , \( 168611 i + 37554\bigr] \)
42250.9-i5 \( \bigl[i\) , \( -i\) , \( i\) , \( -7276 i + 16637\) , \( -766037 i - 429510\bigr] \)
42250.9-i6 \( \bigl[i\) , \( -i\) , \( i\) , \( -52151 i + 52512\) , \( 2113463 i + 6889490\bigr] \)
42250.9-i7 \( \bigl[i\) , \( -i\) , \( i\) , \( 2434 i - 583\) , \( -41589 i - 19046\bigr] \)
42250.9-i8 \( \bigl[1\) , \( i\) , \( 1\) , \( 38504 i - 9074\) , \( 2742317 i + 1271550\bigr] \)