Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-20736.3-p
Number of curves 8
Graph
Conductor 20736.3
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([0,0]),K([0,0]),K([-957,198]),K([11034,-3550])]) 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 20736.3-p have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 12 & 4 & 12 & 6 & 2 & 4 & 3 \\ 12 & 1 & 12 & 4 & 2 & 6 & 3 & 4 \\ 4 & 12 & 1 & 3 & 6 & 2 & 4 & 12 \\ 12 & 4 & 3 & 1 & 2 & 6 & 12 & 4 \\ 6 & 2 & 6 & 2 & 1 & 3 & 6 & 2 \\ 2 & 6 & 2 & 6 & 3 & 1 & 2 & 6 \\ 4 & 3 & 4 & 12 & 6 & 2 & 1 & 12 \\ 3 & 4 & 12 & 4 & 2 & 6 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 20736.3-p over \(\Q(\sqrt{-2}) \)

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

Isogeny class 20736.3-p contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
20736.3-p1 \( \bigl[0\) , \( 0\) , \( 0\) , \( 198 a - 957\) , \( -3550 a + 11034\bigr] \)
20736.3-p2 \( \bigl[0\) , \( 0\) , \( 0\) , \( -198 a - 957\) , \( -3550 a - 11034\bigr] \)
20736.3-p3 \( \bigl[0\) , \( 0\) , \( 0\) , \( 18 a - 12\) , \( 41 a + 9\bigr] \)
20736.3-p4 \( \bigl[0\) , \( 0\) , \( 0\) , \( -18 a - 12\) , \( 41 a - 9\bigr] \)
20736.3-p5 \( \bigl[0\) , \( 0\) , \( 0\) , \( -18 a - 57\) , \( -22 a - 198\bigr] \)
20736.3-p6 \( \bigl[0\) , \( 0\) , \( 0\) , \( 18 a - 57\) , \( -22 a + 198\bigr] \)
20736.3-p7 \( \bigl[0\) , \( 0\) , \( 0\) , \( -162 a + 123\) , \( -526 a + 1458\bigr] \)
20736.3-p8 \( \bigl[0\) , \( 0\) , \( 0\) , \( 162 a + 123\) , \( -526 a - 1458\bigr] \)