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Properties

Base field \(\Q(\sqrt{14}) \)
Label 2.2.56.1-56.1-d
Number of curves 6
Graph
Conductor 56.1
Rank \( 2 \)

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Base field \(\Q(\sqrt{14}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 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 \( x^{2} - 14 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-635,150]),K([-9041,2318])]) 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 56.1-d have rank \( 2 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 8 & 4 & 2 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 8 & 4 & 1 & 2 & 4 & 8 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 2 & 4 & 4 & 2 & 1 & 2 \\ 4 & 8 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 56.1-d over \(\Q(\sqrt{14}) \)

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

Isogeny class 56.1-d contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
56.1-d1 \( \bigl[a\) , \( 1\) , \( a\) , \( 150 a - 635\) , \( 2318 a - 9041\bigr] \)
56.1-d2 \( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \)
56.1-d3 \( \bigl[a\) , \( 1\) , \( a\) , \( -15\) , \( -5\bigr] \)
56.1-d4 \( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( -11\bigr] \)
56.1-d5 \( \bigl[a\) , \( 1\) , \( a\) , \( -75\) , \( -361\bigr] \)
56.1-d6 \( \bigl[a\) , \( 1\) , \( a\) , \( -150 a - 635\) , \( -2318 a - 9041\bigr] \)