Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-7168.7-b
Number of curves 6
Graph
Conductor 7168.7
Rank \( 0 \)

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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([0,0]),K([1,0]),K([0,0]),K([-682,171]),K([6298,-2369])]) 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 7168.7-b have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 7168.7-b over \(\Q(\sqrt{-7}) \)

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

Isogeny class 7168.7-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
7168.7-b1 \( \bigl[0\) , \( 1\) , \( 0\) , \( 171 a - 682\) , \( -2369 a + 6298\bigr] \)
7168.7-b2 \( \bigl[0\) , \( -1\) , \( 0\) , \( -65 a + 94\) , \( -23 a - 422\bigr] \)
7168.7-b3 \( \bigl[0\) , \( 1\) , \( 0\) , \( 11 a - 42\) , \( -33 a + 90\bigr] \)
7168.7-b4 \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a + 16\) , \( -36 a - 20\bigr] \)
7168.7-b5 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -68 a + 16\) , \( 144 a - 368\bigr] \)
7168.7-b6 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 12 a + 16\) , \( 48 a - 48\bigr] \)