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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([-1,0]),K([0,0]),K([-306,136]),K([5425,-4774])]) 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 3087.1-h have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 3087.1-h over \(\Q(\sqrt{2}) \)

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

Isogeny class 3087.1-h contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
3087.1-h1 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 136 a - 306\) , \( -4774 a + 5425\bigr] \)
3087.1-h2 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -4 a + 9\) , \( 0\bigr] \)
3087.1-h3 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 16 a - 36\) , \( -22 a + 25\bigr] \)
3087.1-h4 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 156 a - 351\) , \( 1980 a - 2250\bigr] \)
3087.1-h5 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 196 a - 441\) , \( -2992 a + 3400\bigr] \)
3087.1-h6 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 3136 a - 7056\) , \( -187330 a + 212875\bigr] \)