↑

Properties

Base field \(\Q(\sqrt{15}) \)
Label 2.2.60.1-10.1-c
Number of curves 4
Graph
Conductor 10.1
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{15}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-15, 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 \( x^{2} - 15 \); class number \(2\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([110,-24]),K([-960,250])]) 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 10.1-c have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 10.1-c over \(\Q(\sqrt{15}) \)

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

Isogeny class 10.1-c contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
10.1-c1 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( -24 a + 110\) , \( 250 a - 960\bigr] \)
10.1-c2 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( 16 a - 30\) , \( 42 a - 128\bigr] \)
10.1-c3 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( 196 a - 730\) , \( 2746 a - 10608\bigr] \)
10.1-c4 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( 176 a - 890\) , \( 2490 a - 8960\bigr] \)