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Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-2601.3-a
Number of curves 4
Graph
Conductor 2601.3
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([-378,-119]),K([2952,1220])]) 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 2601.3-a 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 2601.3-a over \(\Q(\sqrt{2}) \)

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

Isogeny class 2601.3-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
2601.3-a1 \( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -119 a - 378\) , \( 1220 a + 2952\bigr] \)
2601.3-a2 \( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( a + 2\) , \( -5 a - 12\bigr] \)
2601.3-a3 \( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -7 a - 16\) , \( -10 a - 34\bigr] \)
2601.3-a4 \( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -487 a - 1536\) , \( 11165 a + 24516\bigr] \)