Properties

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

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

Elliptic curves in class 6720.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.g1 6720a3 \([0, -1, 0, -56481, -5147775]\) \(608119035935048/826875\) \(27095040000\) \([2]\) \(12288\) \(1.2761\)  
6720.g2 6720a4 \([0, -1, 0, -8961, 221121]\) \(2428799546888/778248135\) \(25501634887680\) \([2]\) \(12288\) \(1.2761\)  
6720.g3 6720a2 \([0, -1, 0, -3561, -78039]\) \(1219555693504/43758225\) \(179233689600\) \([2, 2]\) \(6144\) \(0.92955\)  
6720.g4 6720a1 \([0, -1, 0, 84, -4410]\) \(1012048064/130203045\) \(-8332994880\) \([2]\) \(3072\) \(0.58297\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 6720.2.a.g

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