Properties

Label 62475f
Number of curves $4$
Conductor $62475$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 62475f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
62475.cd4 62475f1 \([1, 1, 0, -221750, 91248375]\) \(-656008386769/1581036975\) \(-2906365922996484375\) \([2]\) \(884736\) \(2.2309\) \(\Gamma_0(N)\)-optimal
62475.cd3 62475f2 \([1, 1, 0, -4686875, 3900000000]\) \(6193921595708449/6452105625\) \(11860683979306640625\) \([2, 2]\) \(1769472\) \(2.5775\)  
62475.cd2 62475f3 \([1, 1, 0, -5844500, 1824378375]\) \(12010404962647729/6166198828125\) \(11335111342657470703125\) \([2]\) \(3538944\) \(2.9241\)  
62475.cd1 62475f4 \([1, 1, 0, -74971250, 249825028125]\) \(25351269426118370449/27551475\) \(50646929410546875\) \([2]\) \(3538944\) \(2.9241\)  

Rank

sage: E.rank()
 

The elliptic curves in class 62475f have rank \(0\).

Complex multiplication

The elliptic curves in class 62475f do not have complex multiplication.

Modular form 62475.2.a.f

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