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Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-2025.1-e
Number of curves 8
Graph
Conductor 2025.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{5}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,0]),K([6,-6]),K([10,-7])]) 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 2025.1-e have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 15 & 5 & 3 & 10 & 6 & 2 & 30 \\ 15 & 1 & 3 & 5 & 6 & 10 & 30 & 2 \\ 5 & 3 & 1 & 15 & 2 & 30 & 10 & 6 \\ 3 & 5 & 15 & 1 & 30 & 2 & 6 & 10 \\ 10 & 6 & 2 & 30 & 1 & 15 & 5 & 3 \\ 6 & 10 & 30 & 2 & 15 & 1 & 3 & 5 \\ 2 & 30 & 10 & 6 & 5 & 3 & 1 & 15 \\ 30 & 2 & 6 & 10 & 3 & 5 & 15 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 2025.1-e over \(\Q(\sqrt{5}) \)

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

Isogeny class 2025.1-e contains 8 curves linked by isogenies of degrees dividing 30.

Curve label Weierstrass Coefficients
2025.1-e1 \( \bigl[\phi\) , \( -\phi - 1\) , \( 0\) , \( -6 \phi + 6\) , \( -7 \phi + 10\bigr] \)
2025.1-e2 \( \bigl[\phi\) , \( -\phi - 1\) , \( 1\) , \( -59 \phi + 51\) , \( 253 \phi - 264\bigr] \)
2025.1-e3 \( \bigl[\phi + 1\) , \( 1\) , \( \phi + 1\) , \( 6 \phi - 1\) , \( 13 \phi + 2\bigr] \)
2025.1-e4 \( \bigl[\phi + 1\) , \( 1\) , \( \phi\) , \( 59 \phi - 8\) , \( -194 \phi - 19\bigr] \)
2025.1-e5 \( \bigl[\phi + 1\) , \( 1\) , \( \phi + 1\) , \( 6 \phi - 76\) , \( 178 \phi - 193\bigr] \)
2025.1-e6 \( \bigl[\phi + 1\) , \( 1\) , \( \phi\) , \( 59 \phi - 683\) , \( -5324 \phi + 3896\bigr] \)
2025.1-e7 \( \bigl[\phi\) , \( -\phi - 1\) , \( 0\) , \( -6 \phi - 69\) , \( -172 \phi + 55\bigr] \)
2025.1-e8 \( \bigl[\phi\) , \( -\phi - 1\) , \( 1\) , \( -59 \phi - 624\) , \( 5383 \phi - 804\bigr] \)