Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-10368.3-d
Number of curves 4
Graph
Conductor 10368.3
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([-6,24]),K([44,52])]) 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 10368.3-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 10368.3-d over \(\Q(\sqrt{-2}) \)

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

Isogeny class 10368.3-d contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
10368.3-d1 \( \bigl[0\) , \( 0\) , \( 0\) , \( 24 a - 6\) , \( 52 a + 44\bigr] \)
10368.3-d2 \( \bigl[a\) , \( -1\) , \( 0\) , \( -21 a + 6\) , \( 31 a - 69\bigr] \)
10368.3-d3 \( \bigl[0\) , \( 0\) , \( 0\) , \( 6 a + 21\) , \( -38 a + 26\bigr] \)
10368.3-d4 \( \bigl[a\) , \( -1\) , \( 0\) , \( 24 a + 96\) , \( 229 a - 240\bigr] \)