Properties

Label 135240.o
Number of curves $4$
Conductor $135240$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 135240.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
135240.o1 135240by4 \([0, -1, 0, -8416256, -9395001300]\) \(273629163383866082/26408025\) \(6362885597644800\) \([2]\) \(3932160\) \(2.4662\)  
135240.o2 135240by3 \([0, -1, 0, -932976, 109990476]\) \(372749784765122/194143359375\) \(46777901234400000000\) \([2]\) \(3932160\) \(2.4662\)  
135240.o3 135240by2 \([0, -1, 0, -527256, -145937700]\) \(134555337776164/1312250625\) \(158090213151360000\) \([2, 2]\) \(1966080\) \(2.1196\)  
135240.o4 135240by1 \([0, -1, 0, -8836, -5549564]\) \(-2533446736/440749575\) \(-13274559167788800\) \([4]\) \(983040\) \(1.7731\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 135240.o have rank \(0\).

Complex multiplication

The elliptic curves in class 135240.o do not have complex multiplication.

Modular form 135240.2.a.o

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.