Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-128.6-c
Number of curves 4
Graph
Conductor 128.6
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([1,1]),K([-1,-1]),K([1,1]),K([-12018824,2811452]),K([21298939255,-4982304577])]) 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 128.6-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 128.6-c over \(\Q(\sqrt{57}) \)

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

Isogeny class 128.6-c contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
128.6-c1 \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 2811452 a - 12018824\) , \( -4982304577 a + 21298939255\bigr] \)
128.6-c2 \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -115777 a - 381257\) , \( -42294261 a - 138472621\bigr] \)
128.6-c3 \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -8428 a + 36016\) , \( 733567 a - 3135945\bigr] \)
128.6-c4 \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 353 a + 1153\) , \( 7663 a + 25095\bigr] \)