Properties

Label 6864.d
Number of curves $4$
Conductor $6864$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 6864.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6864.d1 6864c4 \([0, -1, 0, -1704, 27600]\) \(267335955794/570999\) \(1169405952\) \([2]\) \(4096\) \(0.62438\)  
6864.d2 6864c3 \([0, -1, 0, -1464, -20976]\) \(169556172914/942513\) \(1930266624\) \([2]\) \(4096\) \(0.62438\)  
6864.d3 6864c2 \([0, -1, 0, -144, 144]\) \(324730948/184041\) \(188457984\) \([2, 2]\) \(2048\) \(0.27780\)  
6864.d4 6864c1 \([0, -1, 0, 36, 0]\) \(19600688/11583\) \(-2965248\) \([2]\) \(1024\) \(-0.068769\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6864.d have rank \(1\).

Complex multiplication

The elliptic curves in class 6864.d do not have complex multiplication.

Modular form 6864.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} - q^{11} - q^{13} + 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

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

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.