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Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-4053.3-a
Number of curves 6
Graph
Conductor 4053.3
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([1,0]),K([1,1]),K([1,1]),K([-6,16]),K([4,23])]) 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 4053.3-a 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 4053.3-a over \(\Q(\sqrt{-3}) \)

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

Isogeny class 4053.3-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
4053.3-a1 \( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 16 a - 6\) , \( 23 a + 4\bigr] \)
4053.3-a2 \( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -224 a + 54\) , \( -1621 a + 1222\bigr] \)
4053.3-a3 \( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 16 a - 51\) , \( -76 a + 148\bigr] \)
4053.3-a4 \( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 266 a - 941\) , \( -3712 a + 7612\bigr] \)
4053.3-a5 \( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 256 a - 876\) , \( -4147 a + 9190\bigr] \)
4053.3-a6 \( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 4086 a - 14011\) , \( -260566 a + 609796\bigr] \)