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Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-1083.2-b
Number of curves 6
Graph
Conductor 1083.2
Rank \( 0 \)

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Base field \(\Q(\sqrt{-3}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([63,130]),K([999,-464])]) 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 1083.2-b have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrr} 1 & 4 & 2 & 8 & 4 & 8 \\ 4 & 1 & 2 & 8 & 4 & 8 \\ 2 & 2 & 1 & 4 & 2 & 4 \\ 8 & 8 & 4 & 1 & 2 & 4 \\ 4 & 4 & 2 & 2 & 1 & 2 \\ 8 & 8 & 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 1083.2-b over \(\Q(\sqrt{-3}) \)

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

Isogeny class 1083.2-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
1083.2-b1 \( \bigl[1\) , \( 0\) , \( 1\) , \( 130 a + 63\) , \( -464 a + 999\bigr] \)
1083.2-b2 \( \bigl[1\) , \( 0\) , \( 1\) , \( -130 a + 193\) , \( 464 a + 535\bigr] \)
1083.2-b3 \( \bigl[1\) , \( 0\) , \( 1\) , \( 8\) , \( 29\bigr] \)
1083.2-b4 \( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( -1\bigr] \)
1083.2-b5 \( \bigl[1\) , \( 0\) , \( 1\) , \( -7\) , \( 5\bigr] \)
1083.2-b6 \( \bigl[1\) , \( 0\) , \( 1\) , \( -102\) , \( 385\bigr] \)