Properties

Label 69825by
Number of curves $4$
Conductor $69825$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("by1")
 
E.isogeny_class()
 

Elliptic curves in class 69825by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69825.q3 69825by1 \([1, 0, 0, -1863, 26592]\) \(389017/57\) \(104781140625\) \([2]\) \(73728\) \(0.83952\) \(\Gamma_0(N)\)-optimal
69825.q2 69825by2 \([1, 0, 0, -7988, -249033]\) \(30664297/3249\) \(5972525015625\) \([2, 2]\) \(147456\) \(1.1861\)  
69825.q4 69825by3 \([1, 0, 0, 10387, -1222908]\) \(67419143/390963\) \(-718693843546875\) \([2]\) \(294912\) \(1.5327\)  
69825.q1 69825by4 \([1, 0, 0, -124363, -16890658]\) \(115714886617/1539\) \(2829090796875\) \([2]\) \(294912\) \(1.5327\)  

Rank

sage: E.rank()
 

The elliptic curves in class 69825by have rank \(0\).

Complex multiplication

The elliptic curves in class 69825by do not have complex multiplication.

Modular form 69825.2.a.by

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