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Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-31.1-a
Number of curves 4
Graph
Conductor 31.1
Rank \( 0 \)

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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([1,1]),K([-1,1]),K([1,1]),K([-21,-7]),K([-52,-31])]) 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 31.1-a have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 31.1-a over \(\Q(\sqrt{2}) \)

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

Isogeny class 31.1-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
31.1-a1 \( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -7 a - 21\) , \( -31 a - 52\bigr] \)
31.1-a2 \( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -2 a - 1\) , \( -a - 2\bigr] \)
31.1-a3 \( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 10 a - 14\) , \( 21 a - 30\bigr] \)
31.1-a4 \( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 25 a - 49\) , \( -79 a + 100\bigr] \)