Properties

Base field \(\Q(\sqrt{-95}) \)
Label 2.0.95.1-36.1-a
Number of curves 2
Graph
Conductor 36.1
Rank \( 1 \)

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Base field \(\Q(\sqrt{-95}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([24, -1, 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 \( x^{2} - x + 24 \); class number \(8\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([0,3]),K([12,7])]) 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 36.1-a have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 36.1-a over \(\Q(\sqrt{-95}) \)

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

Isogeny class 36.1-a contains 2 curves linked by isogenies of degree 3.

Curve label Weierstrass Coefficients
36.1-a1 \( \bigl[0\) , \( a\) , \( 0\) , \( 3 a\) , \( 7 a + 12\bigr] \)
36.1-a2 \( \bigl[0\) , \( a\) , \( a\) , \( -141 a + 1872\) , \( -5178 a - 12924\bigr] \)