Properties

Label 6720.b
Number of curves $6$
Conductor $6720$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.b1 6720e4 \([0, -1, 0, -26881, 1705345]\) \(32779037733124/315\) \(20643840\) \([2]\) \(8192\) \(0.98191\)  
6720.b2 6720e5 \([0, -1, 0, -25921, -1592255]\) \(14695548366242/57421875\) \(7526400000000\) \([2]\) \(16384\) \(1.3285\)  
6720.b3 6720e3 \([0, -1, 0, -2401, 2401]\) \(23366901604/13505625\) \(885104640000\) \([2, 2]\) \(8192\) \(0.98191\)  
6720.b4 6720e2 \([0, -1, 0, -1681, 27025]\) \(32082281296/99225\) \(1625702400\) \([2, 2]\) \(4096\) \(0.63534\)  
6720.b5 6720e1 \([0, -1, 0, -61, 781]\) \(-24918016/229635\) \(-235146240\) \([2]\) \(2048\) \(0.28876\) \(\Gamma_0(N)\)-optimal
6720.b6 6720e6 \([0, -1, 0, 9599, 9601]\) \(746185003198/432360075\) \(-56670299750400\) \([2]\) \(16384\) \(1.3285\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 6720.2.a.b

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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.