Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-2304.1-l
Number of curves 4
Graph
Conductor 2304.1
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([0,0]),K([1,-1]),K([0,0]),K([-4,-4]),K([26,18])]) 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 2304.1-l 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 2304.1-l over \(\Q(\sqrt{2}) \)

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

Isogeny class 2304.1-l contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
2304.1-l1 \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4 a - 4\) , \( 18 a + 26\bigr] \)
2304.1-l2 \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -24 a - 64\) , \( -36 a + 44\bigr] \)
2304.1-l3 \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -24 a - 34\) , \( 72 a + 104\bigr] \)
2304.1-l4 \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 24 a - 64\) , \( 36 a + 44\bigr] \)