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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([-3, 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 \( x^{2} - 3 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-1143,660]),K([-21034,12144])]) 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 4608.1-a have rank \( 0 \).

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 4608.1-a over \(\Q(\sqrt{3}) \)

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

Isogeny class 4608.1-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
4608.1-a1 \( \bigl[0\) , \( 0\) , \( 0\) , \( 660 a - 1143\) , \( 12144 a - 21034\bigr] \)
4608.1-a2 \( \bigl[0\) , \( 0\) , \( 0\) , \( 45 a - 78\) , \( 153 a - 265\bigr] \)
4608.1-a3 \( \bigl[0\) , \( 0\) , \( 0\) , \( -18 a - 36\) , \( -60 a - 100\bigr] \)
4608.1-a4 \( \bigl[0\) , \( 0\) , \( 0\) , \( -120 a + 207\) , \( 1014 a - 1756\bigr] \)