Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-6272.7-d
Number of curves 6
Graph
Conductor 6272.7
Rank \( 1 \)

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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([1,1]),K([0,0]),K([0,0]),K([150,746]),K([-14440,-424])]) 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 6272.7-d have rank \( 1 \).

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 6272.7-d over \(\Q(\sqrt{-7}) \)

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

Isogeny class 6272.7-d contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
6272.7-d1 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 746 a + 150\) , \( -424 a - 14440\bigr] \)
6272.7-d2 \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -68 a - 129\) , \( 470 a + 555\bigr] \)
6272.7-d3 \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 46 a + 10\) , \( -4 a - 244\bigr] \)
6272.7-d4 \( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -19 a - 7\) , \( 85 a - 27\bigr] \)
6272.7-d5 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 39 a - 174\) , \( -23 a + 980\bigr] \)
6272.7-d6 \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -31 a + 36\) , \( -37 a + 126\bigr] \)