Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-1690.6-a
Number of curves 4
Graph
Conductor 1690.6
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: SageMath

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([1,1]),K([-2,-28]),K([-47,43])]) 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 1690.6-a have rank \( 1 \).

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 1690.6-a over \(\Q(\sqrt{-1}) \)

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

Isogeny class 1690.6-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
1690.6-a1 \( \bigl[1\) , \( -i\) , \( i + 1\) , \( -28 i - 2\) , \( 43 i - 47\bigr] \)
1690.6-a2 \( \bigl[i\) , \( i\) , \( i + 1\) , \( -3 i + 4\) , \( -3 i - 2\bigr] \)
1690.6-a3 \( \bigl[i\) , \( -i\) , \( 0\) , \( 7 i + 1\) , \( 2 i + 8\bigr] \)
1690.6-a4 \( \bigl[1\) , \( i\) , \( 0\) , \( 107 i + 21\) , \( -190 i - 420\bigr] \)