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Properties

Base field \(\Q(\sqrt{-11}) \)
Label 2.0.11.1-27500.3-j
Number of curves 3
Graph
Conductor 27500.3
Rank \( 1 \)

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Base field \(\Q(\sqrt{-11}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -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 \( x^{2} - x + 3 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-30328,0]),K([2020281,0])]) 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 27500.3-j have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrr} 1 & 25 & 5 \\ 25 & 1 & 5 \\ 5 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 27500.3-j over \(\Q(\sqrt{-11}) \)

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

Isogeny class 27500.3-j contains 3 curves linked by isogenies of degrees dividing 25.

Curve label Weierstrass Coefficients
27500.3-j1 \( \bigl[1\) , \( 1\) , \( 1\) , \( -30328\) , \( 2020281\bigr] \)
27500.3-j2 \( \bigl[1\) , \( 1\) , \( 1\) , \( -28\) , \( -69\bigr] \)
27500.3-j3 \( \bigl[1\) , \( 1\) , \( 1\) , \( 197\) , \( 681\bigr] \)