Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("bz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.bz1 6720ba4 \([0, 1, 0, -1296065, 567489663]\) \(7347751505995469192/72930375\) \(2389782528000\) \([2]\) \(61440\) \(1.9515\)  
6720.bz2 6720ba3 \([0, 1, 0, -116065, 425663]\) \(5276930158229192/3050936350875\) \(99973082345472000\) \([2]\) \(61440\) \(1.9515\)  
6720.bz3 6720ba2 \([0, 1, 0, -81065, 8832663]\) \(14383655824793536/45209390625\) \(185177664000000\) \([2, 2]\) \(30720\) \(1.6049\)  
6720.bz4 6720ba1 \([0, 1, 0, -2940, 254538]\) \(-43927191786304/415283203125\) \(-26578125000000\) \([2]\) \(15360\) \(1.2583\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 6720.2.a.bz

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