Properties

Base field \(\Q(\sqrt{15}) \)
Label 2.2.60.1-14.1-a
Number of curves 3
Graph
Conductor 14.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{15}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-15, 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 \( x^{2} - 15 \); class number \(2\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([1,0]),K([-19593,5060]),K([-1524653,393664])]) 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 14.1-a have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 14.1-a over \(\Q(\sqrt{15}) \)

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

Isogeny class 14.1-a contains 3 curves linked by isogenies of degrees dividing 9.

Curve label Weierstrass Coefficients
14.1-a1 \( \bigl[a\) , \( 1\) , \( 1\) , \( 5060 a - 19593\) , \( 393664 a - 1524653\bigr] \)
14.1-a2 \( \bigl[a\) , \( 1\) , \( 1\) , \( 90 a - 343\) , \( 152 a - 585\bigr] \)
14.1-a3 \( \bigl[a\) , \( 1\) , \( 1\) , \( 5010 a - 19398\) , \( -371997 a + 1440742\bigr] \)