Properties

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

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 6720.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.u1 6720bn4 \([0, -1, 0, -14945, 708225]\) \(5633270409316/14175\) \(928972800\) \([2]\) \(8192\) \(0.95905\)  
6720.u2 6720bn3 \([0, -1, 0, -2625, -37023]\) \(30534944836/8203125\) \(537600000000\) \([2]\) \(8192\) \(0.95905\)  
6720.u3 6720bn2 \([0, -1, 0, -945, 11025]\) \(5702413264/275625\) \(4515840000\) \([2, 2]\) \(4096\) \(0.61247\)  
6720.u4 6720bn1 \([0, -1, 0, 35, 637]\) \(4499456/180075\) \(-184396800\) \([2]\) \(2048\) \(0.26590\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 6720.2.a.u

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