Properties

Label 6840.g
Number of curves $4$
Conductor $6840$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6840.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6840.g1 6840r3 \([0, 0, 0, -273603, -55084498]\) \(3034301922374404/1425\) \(1063756800\) \([2]\) \(16384\) \(1.5058\)  
6840.g2 6840r4 \([0, 0, 0, -20523, -491722]\) \(1280615525284/601171875\) \(448772400000000\) \([2]\) \(16384\) \(1.5058\)  
6840.g3 6840r2 \([0, 0, 0, -17103, -860398]\) \(2964647793616/2030625\) \(378963360000\) \([2, 2]\) \(8192\) \(1.1592\)  
6840.g4 6840r1 \([0, 0, 0, -858, -18907]\) \(-5988775936/9774075\) \(-114004810800\) \([4]\) \(4096\) \(0.81261\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6840.g have rank \(0\).

Complex multiplication

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

Modular form 6840.2.a.g

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