Properties

Label 67280.a
Number of curves $4$
Conductor $67280$
CM no
Rank $2$
Graph

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

Elliptic curves in class 67280.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67280.a1 67280q3 \([0, 1, 0, -34761, 2482414]\) \(488095744/125\) \(1189646642000\) \([2]\) \(145152\) \(1.3030\)  
67280.a2 67280q4 \([0, 1, 0, -30556, 3109800]\) \(-20720464/15625\) \(-2379293284000000\) \([2]\) \(290304\) \(1.6496\)  
67280.a3 67280q1 \([0, 1, 0, -1121, -10310]\) \(16384/5\) \(47585865680\) \([2]\) \(48384\) \(0.75370\) \(\Gamma_0(N)\)-optimal
67280.a4 67280q2 \([0, 1, 0, 3084, -65816]\) \(21296/25\) \(-3806869254400\) \([2]\) \(96768\) \(1.1003\)  

Rank

sage: E.rank()
 

The elliptic curves in class 67280.a have rank \(2\).

Complex multiplication

The elliptic curves in class 67280.a do not have complex multiplication.

Modular form 67280.2.a.a

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.