↑

Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-16.1-a
Number of curves 8
Graph
Conductor 16.1
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{3}) \)

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

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([1,0]),K([-5,-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 16.1-a have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 12 & 4 & 6 & 2 & 4 & 12 \\ 3 & 1 & 4 & 12 & 2 & 6 & 12 & 4 \\ 12 & 4 & 1 & 12 & 2 & 6 & 3 & 4 \\ 4 & 12 & 12 & 1 & 6 & 2 & 4 & 3 \\ 6 & 2 & 2 & 6 & 1 & 3 & 6 & 2 \\ 2 & 6 & 6 & 2 & 3 & 1 & 2 & 6 \\ 4 & 12 & 3 & 4 & 6 & 2 & 1 & 12 \\ 12 & 4 & 4 & 3 & 2 & 6 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 16.1-a over \(\Q(\sqrt{3}) \)

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

Isogeny class 16.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
16.1-a1 \( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( -3 a - 5\bigr] \)
16.1-a2 \( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( 3 a + 5\bigr] \)
16.1-a3 \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 4 a - 13\) , \( 11 a - 21\bigr] \)
16.1-a4 \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 4 a - 13\) , \( -12 a + 19\bigr] \)
16.1-a5 \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a - 3\) , \( -1\bigr] \)
16.1-a6 \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -a - 3\) , \( -a - 1\bigr] \)
16.1-a7 \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -6 a - 13\) , \( -12 a - 21\bigr] \)
16.1-a8 \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -6 a - 13\) , \( 11 a + 19\bigr] \)