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Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-22500.1-a
Number of curves 4
Graph
Conductor 22500.1
Rank \( 1 \)

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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([0,0]),K([0,0]),K([192,-192]),K([1216,0])]) 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 22500.1-a have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 22500.1-a over \(\Q(\sqrt{-3}) \)

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

Isogeny class 22500.1-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
22500.1-a1 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -192 a + 192\) , \( 1216\bigr] \)
22500.1-a2 \( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 145 a - 146\) , \( 147\bigr] \)
22500.1-a3 \( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -605 a + 604\) , \( 1647\bigr] \)
22500.1-a4 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -3192 a + 3192\) , \( 70216\bigr] \)