↑

Properties

Base field \(\Q(\sqrt{-11}) \)
Label 2.0.11.1-36864.2-c
Number of curves 4
Graph
Conductor 36864.2
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{-11}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -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 \( x^{2} - x + 3 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([-20,-1]),K([42,3])]) 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 36864.2-c have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 10 & 2 & 5 \\ 10 & 1 & 5 & 2 \\ 2 & 5 & 1 & 10 \\ 5 & 2 & 10 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 36864.2-c over \(\Q(\sqrt{-11}) \)

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

Isogeny class 36864.2-c contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
36864.2-c1 \( \bigl[0\) , \( -1\) , \( 0\) , \( -a - 20\) , \( 3 a + 42\bigr] \)
36864.2-c2 \( \bigl[0\) , \( -1\) , \( 0\) , \( 4 a - 85\) , \( 20 a - 275\bigr] \)
36864.2-c3 \( \bigl[0\) , \( -1\) , \( 0\) , \( 4 a - 25\) , \( 20 a + 25\bigr] \)
36864.2-c4 \( \bigl[0\) , \( -1\) , \( 0\) , \( -a - 5\) , \( 3 a - 3\bigr] \)