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Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-896.4-a
Number of curves 6
Graph
Conductor 896.4
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([0,1]),K([0,0]),K([255,-213]),K([2380,-80])]) 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 896.4-a 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 896.4-a over \(\Q(\sqrt{-7}) \)

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

Isogeny class 896.4-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
896.4-a1 \( \bigl[0\) , \( a\) , \( 0\) , \( -213 a + 255\) , \( -80 a + 2380\bigr] \)
896.4-a2 \( \bigl[0\) , \( -a\) , \( 0\) , \( 40 a - 15\) , \( -74 a - 60\bigr] \)
896.4-a3 \( \bigl[0\) , \( a\) , \( 0\) , \( -13 a + 15\) , \( 44\bigr] \)
896.4-a4 \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 5 a - 6\) , \( -16 a + 4\bigr] \)
896.4-a5 \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 21 a + 26\) , \( 8 a - 128\bigr] \)
896.4-a6 \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( a - 14\) , \( 12 a - 24\bigr] \)