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Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-49539.3-b
Number of curves 6
Graph
Conductor 49539.3
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([0,1]),K([-1,-1]),K([1,0]),K([-49279,28234]),K([3674293,-3119976])]) 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 49539.3-b have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 49539.3-b over \(\Q(\sqrt{-3}) \)

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

Isogeny class 49539.3-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
49539.3-b1 \( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 28234 a - 49279\) , \( -3119976 a + 3674293\bigr] \)
49539.3-b2 \( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -206 a + 131\) , \( -3636 a + 3211\bigr] \)
49539.3-b3 \( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 1819 a - 3109\) , \( -46890 a + 58129\bigr] \)
49539.3-b4 \( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 7804 a - 8779\) , \( 283860 a - 132383\bigr] \)
49539.3-b5 \( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -11 a + 211\) , \( -1350 a + 455\bigr] \)
49539.3-b6 \( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -5351 a + 2091\) , \( -113886 a + 125417\bigr] \)