Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-42250.6-l
Number of curves 4
Graph
Conductor 42250.6
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([1,1]),K([0,0]),K([-1570,-835]),K([23386,19373])]) 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 42250.6-l have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 2 & 10 & 5 \\ 2 & 1 & 5 & 10 \\ 10 & 5 & 1 & 2 \\ 5 & 10 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 42250.6-l over \(\Q(\sqrt{-1}) \)

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

Isogeny class 42250.6-l contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
42250.6-l1 \( \bigl[i\) , \( i + 1\) , \( 0\) , \( -835 i - 1570\) , \( 19373 i + 23386\bigr] \)
42250.6-l2 \( \bigl[i\) , \( i + 1\) , \( 0\) , \( -290 i - 5\) , \( -1350 i + 2175\bigr] \)
42250.6-l3 \( \bigl[1\) , \( 0\) , \( 0\) , \( 152 i + 426\) , \( -3008 i + 1636\bigr] \)
42250.6-l4 \( \bigl[1\) , \( 0\) , \( 0\) , \( 2332 i + 6686\) , \( -200700 i + 111192\bigr] \)