↑

Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-171.1-c
Number of curves 3
Graph
Conductor 171.1
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: SageMath

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([0,0]),K([1,1]),K([1,0]),K([-302343,-92319]),K([-97687237,-29828922])]) 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 171.1-c have rank \( 1 \).

Isogeny matrix

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 171.1-c over \(\Q(\sqrt{57}) \)

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

Isogeny class 171.1-c contains 3 curves linked by isogenies of degrees dividing 9.

Curve label Weierstrass Coefficients
171.1-c1 \( \bigl[0\) , \( a + 1\) , \( 1\) , \( -92319 a - 302343\) , \( -29828922 a - 97687237\bigr] \)
171.1-c2 \( \bigl[0\) , \( a + 1\) , \( 1\) , \( -1119 a - 3663\) , \( -44142 a - 144562\bigr] \)
171.1-c3 \( \bigl[0\) , \( a + 1\) , \( 1\) , \( 81 a + 267\) , \( 108 a + 353\bigr] \)