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Properties

Base field \(\Q(\sqrt{-42}) \)
Label 2.0.168.1-14.1-c
Number of curves 6
Graph
Conductor 14.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{-42}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([42, 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 \( x^{2} + 42 \); class number \(4\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-631,0]),K([9015,0])]) 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-c have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrr} 1 & 9 & 3 & 6 & 18 & 2 \\ 9 & 1 & 3 & 6 & 2 & 18 \\ 3 & 3 & 1 & 2 & 6 & 6 \\ 6 & 6 & 2 & 1 & 3 & 3 \\ 18 & 2 & 6 & 3 & 1 & 9 \\ 2 & 18 & 6 & 3 & 9 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 14.1-c over \(\Q(\sqrt{-42}) \)

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

Isogeny class 14.1-c contains 6 curves linked by isogenies of degrees dividing 18.

Curve label Weierstrass Coefficients
14.1-c1 \( \bigl[a\) , \( 1\) , \( a\) , \( -631\) , \( 9015\bigr] \)
14.1-c2 \( \bigl[a\) , \( 1\) , \( a\) , \( 49\) , \( -17\bigr] \)
14.1-c3 \( \bigl[a\) , \( 1\) , \( a\) , \( 69\) , \( -29\bigr] \)
14.1-c4 \( \bigl[a\) , \( 1\) , \( a\) , \( -91\) , \( 963\bigr] \)
14.1-c5 \( \bigl[a\) , \( 1\) , \( a\) , \( 9\) , \( 7\bigr] \)
14.1-c6 \( \bigl[a\) , \( 1\) , \( a\) , \( -10871\) , \( 473911\bigr] \)