↑

Properties

Base field \(\Q(\sqrt{-21}) \)
Label 2.0.84.1-507.1-a
Number of curves 4
Graph
Conductor 507.1
Rank \( 2 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{-21}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([21, 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 \( x^{2} + 21 \); class number \(4\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([4,0]),K([6,0])]) 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 507.1-a have rank \( 2 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 507.1-a over \(\Q(\sqrt{-21}) \)

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

Isogeny class 507.1-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
507.1-a1 \( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \)
507.1-a2 \( \bigl[1\) , \( -1\) , \( 1\) , \( -41\) , \( 96\bigr] \)
507.1-a3 \( \bigl[1\) , \( -1\) , \( 1\) , \( -176\) , \( -768\bigr] \)
507.1-a4 \( \bigl[1\) , \( -1\) , \( 1\) , \( -626\) , \( 6180\bigr] \)