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([1,0]),K([0,0]),K([-53505,30890]),K([6718827,-3879118])]) 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 3072.1-bt have rank \( 0 \).

Isogeny matrix

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

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 8 & 4 & 8 & 2 & 16 & 4 \\ 16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 \\ 8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 \\ 2 & 8 & 4 & 2 & 4 & 1 & 8 & 2 \\ 16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 \\ 4 & 16 & 8 & 4 & 8 & 2 & 16 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 3072.1-bt over \(\Q(\sqrt{3}) \)

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

Isogeny class 3072.1-bt contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
3072.1-bt1 \( \bigl[0\) , \( 1\) , \( 0\) , \( 30890 a - 53505\) , \( -3879118 a + 6718827\bigr] \)
3072.1-bt2 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 32 a + 64\) , \( 1080 a + 1800\bigr] \)
3072.1-bt3 \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -3882 a + 6724\) , \( -217892 a + 377400\bigr] \)
3072.1-bt4 \( \bigl[0\) , \( 1\) , \( 0\) , \( 130 a - 225\) , \( -750 a + 1299\bigr] \)
3072.1-bt5 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -48 a - 96\) , \( 216 a + 360\bigr] \)
3072.1-bt6 \( \bigl[0\) , \( 1\) , \( 0\) , \( 1930 a - 3345\) , \( -60126 a + 104139\bigr] \)
3072.1-bt7 \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -768 a - 1536\) , \( 16632 a + 27720\bigr] \)
3072.1-bt8 \( \bigl[0\) , \( 1\) , \( 0\) , \( 1770 a - 3105\) , \( -69998 a + 121131\bigr] \)