↑

Properties

Base field \(\Q(\sqrt{-217}) \)
Label 2.0.868.1-62.1-b
Number of curves 4
Graph
Conductor 62.1
Rank \( 2 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{-217}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([217, 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 \(a\), with minimal polynomial \( x^{2} + 217 \); class number \(8\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([1020,0]),K([-4565,0])]) 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 62.1-b have rank \( 2 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 62.1-b over \(\Q(\sqrt{-217}) \)

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

Isogeny class 62.1-b contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
62.1-b1 \( \bigl[a\) , \( 1\) , \( a\) , \( 1020\) , \( -4565\bigr] \)
62.1-b2 \( \bigl[a\) , \( 1\) , \( a\) , \( -450\) , \( 20327\bigr] \)
62.1-b3 \( \bigl[a\) , \( 1\) , \( a\) , \( 40\) , \( 24835\bigr] \)
62.1-b4 \( \bigl[a\) , \( 1\) , \( a\) , \( -15150\) , \( 1075983\bigr] \)