Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-28224.1-d
Number of curves 8
Graph
Conductor 28224.1
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([0,1]),K([-1,1]),K([0,0]),K([476,1190]),K([-94178,7595])]) 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 28224.1-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 28224.1-d over \(\Q(\sqrt{-7}) \)

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

Isogeny class 28224.1-d contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
28224.1-d1 \( \bigl[a\) , \( a - 1\) , \( 0\) , \( 1190 a + 476\) , \( 7595 a - 94178\bigr] \)
28224.1-d2 \( \bigl[a\) , \( a - 1\) , \( 0\) , \( -35 a - 14\) , \( 0\bigr] \)
28224.1-d3 \( \bigl[a\) , \( a - 1\) , \( 0\) , \( 140 a + 56\) , \( 35 a - 434\bigr] \)
28224.1-d4 \( \bigl[a\) , \( 1\) , \( a\) , \( 40 a - 162\) , \( 275 a - 795\bigr] \)
28224.1-d5 \( \bigl[a\) , \( a - 1\) , \( 0\) , \( 1365 a + 546\) , \( -3150 a + 39060\bigr] \)
28224.1-d6 \( \bigl[a\) , \( a\) , \( a\) , \( -86 a + 145\) , \( -130 a - 556\bigr] \)
28224.1-d7 \( \bigl[a\) , \( a - 1\) , \( 0\) , \( 1715 a + 686\) , \( 4760 a - 59024\bigr] \)
28224.1-d8 \( \bigl[a\) , \( a - 1\) , \( 0\) , \( 27440 a + 10976\) , \( 298025 a - 3695510\bigr] \)