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Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-1458.1-x
Number of curves 2
Graph
Conductor 1458.1
Rank \( 1 \)

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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([1,0]),K([-1,0]),K([1,0]),K([-32,-18]),K([239,138])]) 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 1458.1-x have rank \( 1 \).

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 1458.1-x over \(\Q(\sqrt{3}) \)

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

Isogeny class 1458.1-x contains 2 curves linked by isogenies of degree 3.

Curve label Weierstrass Coefficients
1458.1-x1 \( \bigl[1\) , \( -1\) , \( 1\) , \( -18 a - 32\) , \( 138 a + 239\bigr] \)
1458.1-x2 \( \bigl[a\) , \( 0\) , \( 0\) , \( 18 a + 30\) , \( -108 a - 188\bigr] \)