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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([-1,0]),K([0,0]),K([-238,714]),K([-19747,-13671])]) 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 7056.5-c have rank \( 0 \).

Isogeny matrix

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

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

Isogeny graph

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

Elliptic curves in class 7056.5-c over \(\Q(\sqrt{-7}) \)

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

Isogeny class 7056.5-c contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
7056.5-c1 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 714 a - 238\) , \( -13671 a - 19747\bigr] \)
7056.5-c2 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -21 a + 7\) , \( 0\bigr] \)
7056.5-c3 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 84 a - 28\) , \( -63 a - 91\bigr] \)
7056.5-c4 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 819 a - 273\) , \( 5670 a + 8190\bigr] \)
7056.5-c5 \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -22 a + 82\) , \( -169 a - 61\bigr] \)
7056.5-c6 \( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -7 a - 74\) , \( -77 a - 197\bigr] \)
7056.5-c7 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 1029 a - 343\) , \( -8568 a - 12376\bigr] \)
7056.5-c8 \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 16464 a - 5488\) , \( -536445 a - 774865\bigr] \)