Properties

Label 6240a
Number of curves $4$
Conductor $6240$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6240a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6240.d3 6240a1 \([0, -1, 0, -66, 216]\) \(504358336/38025\) \(2433600\) \([2, 2]\) \(768\) \(-0.029371\) \(\Gamma_0(N)\)-optimal
6240.d2 6240a2 \([0, -1, 0, -216, -924]\) \(2186875592/428415\) \(219348480\) \([2]\) \(1536\) \(0.31720\)  
6240.d1 6240a3 \([0, -1, 0, -1041, 13281]\) \(30488290624/195\) \(798720\) \([2]\) \(1536\) \(0.31720\)  
6240.d4 6240a4 \([0, -1, 0, 64, 840]\) \(55742968/658125\) \(-336960000\) \([2]\) \(1536\) \(0.31720\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6240a have rank \(1\).

Complex multiplication

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

Modular form 6240.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.