↑

Properties

Base field \(\Q(\sqrt{-85}) \)
Label 2.0.340.1-64.1-a
Number of curves 4
Graph
Conductor 64.1
Rank bounds 1...3

Related objects

Downloads

Learn more

Show commands: SageMath

Base field \(\Q(\sqrt{-85}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([85, 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 \( x^{2} + 85 \); class number \(4\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-49,-12]),K([0,0])]) E.isogeny_class()
 

Rank

Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 

The rank \(r\) of the elliptic curves in class 64.1-a satisfy \(1 \le r \le 3\).

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 64.1-a over \(\Q(\sqrt{-85}) \)

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

Isogeny class 64.1-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
64.1-a1 \( \bigl[0\) , \( 0\) , \( 0\) , \( -12 a - 49\) , \( 0\bigr] \)
64.1-a2 \( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \)
64.1-a3 \( \bigl[a + 1\) , \( a\) , \( 0\) , \( 12 a + 218\) , \( -141 a - 4542\bigr] \)
64.1-a4 \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 11 a + 260\) , \( -61 a + 78\bigr] \)