Properties

Label 13650e
Number of curves $4$
Conductor $13650$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 13650e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13650.b3 13650e1 \([1, 1, 0, -175, -875]\) \(38272753/4368\) \(68250000\) \([2]\) \(6144\) \(0.23554\) \(\Gamma_0(N)\)-optimal
13650.b2 13650e2 \([1, 1, 0, -675, 5625]\) \(2181825073/298116\) \(4658062500\) \([2, 2]\) \(12288\) \(0.58212\)  
13650.b1 13650e3 \([1, 1, 0, -10425, 405375]\) \(8020417344913/187278\) \(2926218750\) \([2]\) \(24576\) \(0.92869\)  
13650.b4 13650e4 \([1, 1, 0, 1075, 31875]\) \(8780064047/32388174\) \(-506065218750\) \([2]\) \(24576\) \(0.92869\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13650e have rank \(2\).

Complex multiplication

The elliptic curves in class 13650e do not have complex multiplication.

Modular form 13650.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - 4 q^{11} - q^{12} + q^{13} + q^{14} + q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.