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Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-128.5-c
Number of curves 4
Graph
Conductor 128.5
Rank \( 0 \)

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Base field \(\Q(\sqrt{57}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -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 \( x^{2} - x - 14 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([0,1]),K([0,1]),K([-497049,115785]),K([-178648906,41797219])]) 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 128.5-c have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 128.5-c over \(\Q(\sqrt{57}) \)

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

Isogeny class 128.5-c contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
128.5-c1 \( \bigl[a\) , \( a\) , \( a\) , \( 115785 a - 497049\) , \( 41797219 a - 178648906\bigr] \)
128.5-c2 \( \bigl[a\) , \( -1\) , \( a\) , \( -2811454 a - 9207370\) , \( 4982304576 a + 16316634679\bigr] \)
128.5-c3 \( \bigl[a\) , \( a\) , \( a\) , \( -345 a + 1491\) , \( -6165 a + 26374\bigr] \)
128.5-c4 \( \bigl[a\) , \( -1\) , \( a\) , \( 8426 a + 27590\) , \( -733568 a - 2402377\bigr] \)