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Properties

Label 6840.d
Number of curves $6$
Conductor $6840$
CM no
Rank $0$
Graph

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

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 6840.2.a.d

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

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 6840.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6840.d1 6840e5 \([0, 0, 0, -6238083, 5996872798]\) \(17981241677724245762/16245\) \(24253655040\) \([2]\) \(65536\) \(2.1879\)  
6840.d2 6840e4 \([0, 0, 0, -389883, 93699718]\) \(8780093172522724/263900025\) \(197000313062400\) \([2, 2]\) \(32768\) \(1.8413\)  
6840.d3 6840e6 \([0, 0, 0, -373683, 101841838]\) \(-3865238121540962/764260336845\) \(-1141034568826890240\) \([2]\) \(65536\) \(2.1879\)  
6840.d4 6840e3 \([0, 0, 0, -110883, -12908882]\) \(201971983086724/20447192475\) \(15263747393817600\) \([2]\) \(32768\) \(1.8413\)  
6840.d5 6840e2 \([0, 0, 0, -25383, 1335418]\) \(9691367618896/1480325625\) \(276264289440000\) \([2, 2]\) \(16384\) \(1.4948\)  
6840.d6 6840e1 \([0, 0, 0, 2742, 114793]\) \(195469297664/601171875\) \(-7012068750000\) \([2]\) \(8192\) \(1.1482\) \(\Gamma_0(N)\)-optimal