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Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-61731.3-b
Number of curves 6
Graph
Conductor 61731.3
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([1,1]),K([-1,-1]),K([0,1]),K([10240,-9143]),K([329450,68164])]) 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 61731.3-b have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 61731.3-b over \(\Q(\sqrt{-3}) \)

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

Isogeny class 61731.3-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
61731.3-b1 \( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -9143 a + 10240\) , \( 68164 a + 329450\bigr] \)
61731.3-b2 \( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -128 a + 535\) , \( -4787 a + 13769\bigr] \)
61731.3-b3 \( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 22 a - 95\) , \( 103 a - 271\bigr] \)
61731.3-b4 \( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 97 a - 410\) , \( -980 a + 2978\bigr] \)
61731.3-b5 \( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 5287 a + 5950\) , \( -322106 a + 404732\bigr] \)
61731.3-b6 \( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 1522 a - 6395\) , \( -66245 a + 194783\bigr] \)