Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-28224.7-d
Number of curves 8
Graph
Conductor 28224.7
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([0,1]),K([0,0]),K([1665,-1190]),K([-84202,-7119])]) 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 28224.7-d have rank \( 1 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 28224.7-d over \(\Q(\sqrt{-7}) \)

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

Isogeny class 28224.7-d contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
28224.7-d1 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( -1190 a + 1665\) , \( -7119 a - 84202\bigr] \)
28224.7-d2 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( 35 a - 50\) , \( -14 a - 69\bigr] \)
28224.7-d3 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( -140 a + 195\) , \( 21 a - 118\bigr] \)
28224.7-d4 \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 86 a + 58\) , \( 274 a - 858\bigr] \)
28224.7-d5 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( -1365 a + 1910\) , \( 3696 a + 38641\bigr] \)
28224.7-d6 \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -42 a - 122\) , \( -276 a - 520\bigr] \)
28224.7-d7 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( -1715 a + 2400\) , \( -4074 a - 50833\bigr] \)
28224.7-d8 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( -27440 a + 38415\) , \( -287049 a - 3342604\bigr] \)