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Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-4.1-d
Number of curves 4
Graph
Conductor 4.1
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([1,0]),K([0,1]),K([0,1]),K([-2104558,-642629]),K([-1787435506,-545795629])]) 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 4.1-d 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 4.1-d over \(\Q(\sqrt{57}) \)

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

Isogeny class 4.1-d contains 4 curves linked by isogenies of degrees dividing 21.

Curve label Weierstrass Coefficients
4.1-d1 \( \bigl[1\) , \( a\) , \( a\) , \( -642629 a - 2104558\) , \( -545795629 a - 1787435506\bigr] \)
4.1-d2 \( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 642628 a - 2747187\) , \( 545795628 a - 2333231135\bigr] \)
4.1-d3 \( \bigl[1\) , \( a\) , \( a\) , \( -239 a - 778\) , \( -6021 a - 19722\bigr] \)
4.1-d4 \( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 238 a - 1017\) , \( 6020 a - 25743\bigr] \)