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Properties

Base field \(\Q(\sqrt{-129}) \)
Label 2.0.516.1-98.2-d
Number of curves 6
Graph
Conductor 98.2
Rank \( 2 \)

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Base field \(\Q(\sqrt{-129}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([129, 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 \( x^{2} + 129 \); class number \(12\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([-1188,0]),K([-7952,0])]) 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 98.2-d have rank \( 2 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrr} 1 & 9 & 3 & 6 & 18 & 2 \\ 9 & 1 & 3 & 6 & 2 & 18 \\ 3 & 3 & 1 & 2 & 6 & 6 \\ 6 & 6 & 2 & 1 & 3 & 3 \\ 18 & 2 & 6 & 3 & 1 & 9 \\ 2 & 18 & 6 & 3 & 9 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 98.2-d over \(\Q(\sqrt{-129}) \)

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

Isogeny class 98.2-d contains 6 curves linked by isogenies of degrees dividing 18.

Curve label Weierstrass Coefficients
98.2-d1 \( \bigl[a\) , \( 0\) , \( 0\) , \( -1188\) , \( -7952\bigr] \)
98.2-d2 \( \bigl[a\) , \( 0\) , \( 0\) , \( 342\) , \( -1184\bigr] \)
98.2-d3 \( \bigl[a\) , \( 0\) , \( 0\) , \( 387\) , \( -1841\bigr] \)
98.2-d4 \( \bigl[a\) , \( 0\) , \( 0\) , \( 27\) , \( 391\bigr] \)
98.2-d5 \( \bigl[a\) , \( 0\) , \( 0\) , \( 252\) , \( 130\bigr] \)
98.2-d6 \( \bigl[a\) , \( 0\) , \( 0\) , \( -24228\) , \( -1219856\bigr] \)