Properties

Label 6840.e
Number of curves $4$
Conductor $6840$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6840.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6840.e1 6840m3 \([0, 0, 0, -21963, -1252762]\) \(784767874322/35625\) \(53187840000\) \([2]\) \(12288\) \(1.1342\)  
6840.e2 6840m4 \([0, 0, 0, -6843, 201782]\) \(23735908082/1954815\) \(2918523156480\) \([2]\) \(12288\) \(1.1342\)  
6840.e3 6840m2 \([0, 0, 0, -1443, -17458]\) \(445138564/81225\) \(60634137600\) \([2, 2]\) \(6144\) \(0.78768\)  
6840.e4 6840m1 \([0, 0, 0, 177, -1582]\) \(3286064/7695\) \(-1436071680\) \([2]\) \(3072\) \(0.44110\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6840.e have rank \(1\).

Complex multiplication

The elliptic curves in class 6840.e do not have complex multiplication.

Modular form 6840.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{11} + 2 q^{13} - 6 q^{17} - q^{19} + O(q^{20})\) 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.