↑

Properties

Base field \(\Q(\sqrt{-10}) \)
Label 2.0.40.1-121.2-b
Number of curves 3
Graph
Conductor 121.2
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{-10}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([10, 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 \( x^{2} + 10 \); class number \(2\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([-31281,0]),K([2139919,0])]) 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 121.2-b have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 121.2-b over \(\Q(\sqrt{-10}) \)

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

Isogeny class 121.2-b contains 3 curves linked by isogenies of degrees dividing 25.

Curve label Weierstrass Coefficients
121.2-b1 \( \bigl[0\) , \( -1\) , \( 0\) , \( -31281\) , \( 2139919\bigr] \)
121.2-b2 \( \bigl[0\) , \( -1\) , \( 0\) , \( -41\) , \( 199\bigr] \)
121.2-b3 \( \bigl[0\) , \( -1\) , \( 0\) , \( -1\) , \( -1\bigr] \)