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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([1,-1]),K([0,0]),K([-24,12]),K([-180,108])]) 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 1536.1-r have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 1536.1-r over \(\Q(\sqrt{3}) \)

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

Isogeny class 1536.1-r contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
1536.1-r1 \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 12 a - 24\) , \( 108 a - 180\bigr] \)
1536.1-r2 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 59058 a - 102289\) , \( 10267037 a - 17783029\bigr] \)
1536.1-r3 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 3693 a - 6394\) , \( 159632 a - 276490\bigr] \)
1536.1-r4 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4758 a - 8239\) , \( 59237 a - 102601\bigr] \)