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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([0,1]),K([0,1]),K([-15,8]),K([16,-26])]) 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 3456.2-d 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 3456.2-d over \(\Q(\sqrt{-2}) \)

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

Isogeny class 3456.2-d contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
3456.2-d1 \( \bigl[a\) , \( a\) , \( a\) , \( 8 a - 15\) , \( -26 a + 16\bigr] \)
3456.2-d2 \( \bigl[a\) , \( a\) , \( a\) , \( 13 a + 5\) , \( 9 a + 21\bigr] \)
3456.2-d3 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a - 2\) , \( 2 a - 10\bigr] \)
3456.2-d4 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -6 a + 3\) , \( -3 a + 15\bigr] \)