Properties

Label 6720.bb
Number of curves $8$
Conductor $6720$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("6720.bb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.bb1 6720l7 [0, -1, 0, -412865, 64377537] [4] 110592  
6720.bb2 6720l4 [0, -1, 0, -368705, 86295105] [4] 36864  
6720.bb3 6720l6 [0, -1, 0, -172865, -26870463] [2, 2] 55296  
6720.bb4 6720l3 [0, -1, 0, -171585, -27299775] [2] 27648  
6720.bb5 6720l2 [0, -1, 0, -23105, 1346625] [2, 2] 18432  
6720.bb6 6720l5 [0, -1, 0, -5185, 3364417] [2] 36864  
6720.bb7 6720l1 [0, -1, 0, -2625, -17343] [2] 9216 \(\Gamma_0(N)\)-optimal
6720.bb8 6720l8 [0, -1, 0, 46655, -90662975] [2] 110592  

Rank

sage: E.rank()
 

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

Modular form 6720.2.a.bb

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.