Properties

Label 6240.p
Number of curves $4$
Conductor $6240$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6240.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6240.p1 6240z2 \([0, -1, 0, -227520, -41682600]\) \(2543984126301795848/909361981125\) \(465593334336000\) \([2]\) \(49152\) \(1.7838\)  
6240.p2 6240z3 \([0, -1, 0, -117520, 15226900]\) \(350584567631475848/8259273550125\) \(4228748057664000\) \([4]\) \(49152\) \(1.7838\)  
6240.p3 6240z1 \([0, -1, 0, -16270, -446600]\) \(7442744143086784/2927948765625\) \(187388721000000\) \([2, 2]\) \(24576\) \(1.4372\) \(\Gamma_0(N)\)-optimal
6240.p4 6240z4 \([0, -1, 0, 52175, -3280223]\) \(3834800837445824/3342041015625\) \(-13689000000000000\) \([4]\) \(49152\) \(1.7838\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6240.p have rank \(0\).

Complex multiplication

The elliptic curves in class 6240.p do not have complex multiplication.

Modular form 6240.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + 4q^{7} + q^{9} + q^{13} - q^{15} + 2q^{17} + 8q^{19} + O(q^{20})\)  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.