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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,-1]),K([1,0]),K([54742,62563]),K([-14087030,9543612])]) 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 20449.3-a 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 20449.3-a over \(\Q(\sqrt{-3}) \)

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

Isogeny class 20449.3-a contains 3 curves linked by isogenies of degrees dividing 25.

Curve label Weierstrass Coefficients
20449.3-a1 \( \bigl[0\) , \( -a\) , \( 1\) , \( 62563 a + 54742\) , \( 9543612 a - 14087030\bigr] \)
20449.3-a2 \( \bigl[0\) , \( -a\) , \( 1\) , \( 83 a + 72\) , \( 782 a - 1200\bigr] \)
20449.3-a3 \( \bigl[0\) , \( -a\) , \( 1\) , \( 3 a + 2\) , \( -8 a + 10\bigr] \)