Properties

Label 6720.x
Number of curves $4$
Conductor $6720$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 6720.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.x1 6720o3 \([0, -1, 0, -1205, -14475]\) \(189123395584/16078125\) \(16464000000\) \([2]\) \(6912\) \(0.70178\)  
6720.x2 6720o1 \([0, -1, 0, -245, 1557]\) \(1594753024/4725\) \(4838400\) \([2]\) \(2304\) \(0.15247\) \(\Gamma_0(N)\)-optimal
6720.x3 6720o2 \([0, -1, 0, -145, 2737]\) \(-20720464/178605\) \(-2926264320\) \([2]\) \(4608\) \(0.49904\)  
6720.x4 6720o4 \([0, -1, 0, 1295, -68975]\) \(14647977776/132355125\) \(-2168506368000\) \([2]\) \(13824\) \(1.0483\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6720.x have rank \(1\).

Complex multiplication

The elliptic curves in class 6720.x do not have complex multiplication.

Modular form 6720.2.a.x

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{7} + q^{9} - 6 q^{11} + 4 q^{13} - q^{15} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.