↑

Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-513.1-a
Number of curves 2
Graph
Conductor 513.1
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{-3}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 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 \( x^{2} - x + 1 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([-21,9]),K([35,-22])]) 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 513.1-a have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 513.1-a over \(\Q(\sqrt{-3}) \)

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

Isogeny class 513.1-a contains 2 curves linked by isogenies of degree 3.

Curve label Weierstrass Coefficients
513.1-a1 \( \bigl[0\) , \( 0\) , \( a\) , \( 9 a - 21\) , \( -22 a + 35\bigr] \)
513.1-a2 \( \bigl[0\) , \( 0\) , \( a\) , \( -a - 1\) , \( -a\bigr] \)