↑

Properties

Base field \(\Q(\sqrt{-182}) \)
Label 2.0.728.1-26.1-c
Number of curves 3
Graph
Conductor 26.1
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{-182}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([182, 0, 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 \( x^{2} + 182 \); class number \(12\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-460,0]),K([-3830,0])]) 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 26.1-c have rank \( 0 \).

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 26.1-c over \(\Q(\sqrt{-182}) \)

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

Isogeny class 26.1-c contains 3 curves linked by isogenies of degrees dividing 9.

Curve label Weierstrass Coefficients
26.1-c1 \( \bigl[1\) , \( 0\) , \( 1\) , \( -460\) , \( -3830\bigr] \)
26.1-c2 \( \bigl[1\) , \( 0\) , \( 1\) , \( -5\) , \( -8\bigr] \)
26.1-c3 \( \bigl[1\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \)