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Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-4761.2-b
Number of curves 4
Graph
Conductor 4761.2
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([-1,-1]),K([1,1]),K([-538,199]),K([-3981,4108])]) 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 4761.2-b have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 5 & 10 & 2 \\ 5 & 1 & 2 & 10 \\ 10 & 2 & 1 & 5 \\ 2 & 10 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 4761.2-b over \(\Q(\sqrt{2}) \)

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

Isogeny class 4761.2-b contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
4761.2-b1 \( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 199 a - 538\) , \( 4108 a - 3981\bigr] \)
4761.2-b2 \( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -a + 2\) , \( -17 a + 15\bigr] \)
4761.2-b3 \( \bigl[a\) , \( a\) , \( a + 1\) , \( 9 a - 25\) , \( -43 a + 37\bigr] \)
4761.2-b4 \( \bigl[a\) , \( a\) , \( a + 1\) , \( 809 a - 2185\) , \( 30112 a - 30333\bigr] \)