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Base field \(\Q(\sqrt{-7}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([-68,-170]),K([1302,-3689])]) 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 4032.1-c have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 8 & 4 & 8 & 16 & 16 & 2 & 4 \\ 8 & 1 & 2 & 4 & 2 & 2 & 4 & 8 \\ 4 & 2 & 1 & 2 & 4 & 4 & 2 & 4 \\ 8 & 4 & 2 & 1 & 8 & 8 & 4 & 8 \\ 16 & 2 & 4 & 8 & 1 & 4 & 8 & 16 \\ 16 & 2 & 4 & 8 & 4 & 1 & 8 & 16 \\ 2 & 4 & 2 & 4 & 8 & 8 & 1 & 2 \\ 4 & 8 & 4 & 8 & 16 & 16 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 4032.1-c over \(\Q(\sqrt{-7}) \)

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

Isogeny class 4032.1-c contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
4032.1-c1 \( \bigl[a\) , \( 1\) , \( 0\) , \( -170 a - 68\) , \( -3689 a + 1302\bigr] \)
4032.1-c2 \( \bigl[a\) , \( 1\) , \( 0\) , \( 5 a + 2\) , \( 0\bigr] \)
4032.1-c3 \( \bigl[a\) , \( 1\) , \( 0\) , \( -20 a - 8\) , \( -17 a + 6\bigr] \)
4032.1-c4 \( \bigl[a\) , \( 1\) , \( 0\) , \( -195 a - 78\) , \( 1530 a - 540\bigr] \)
4032.1-c5 \( \bigl[a\) , \( a - 1\) , \( a\) , \( 11 a - 20\) , \( -38 a + 27\bigr] \)
4032.1-c6 \( \bigl[a\) , \( -1\) , \( 0\) , \( -6 a + 24\) , \( -21 a - 18\bigr] \)
4032.1-c7 \( \bigl[a\) , \( 1\) , \( 0\) , \( -245 a - 98\) , \( -2312 a + 816\bigr] \)
4032.1-c8 \( \bigl[a\) , \( 1\) , \( 0\) , \( -3920 a - 1568\) , \( -144755 a + 51090\bigr] \)