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Properties

Label 197106.by
Number of curves $3$
Conductor $197106$
CM no
Rank $0$
Graph

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

Elliptic curves in class 197106.by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
197106.by1 197106bf3 \([1, 1, 1, -9406404, 11100365253]\) \(-1956469094246217097/36641439744\) \(-1723828813864894464\) \([]\) \(12282192\) \(2.6237\)  
197106.by2 197106bf2 \([1, 1, 1, -43869, 33848883]\) \(-198461344537/10417365504\) \(-490094137834689024\) \([]\) \(4094064\) \(2.0744\)  
197106.by3 197106bf1 \([1, 1, 1, 4866, -1240317]\) \(270840023/14329224\) \(-674130967126344\) \([]\) \(1364688\) \(1.5251\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 197106.by have rank \(0\).

Complex multiplication

The elliptic curves in class 197106.by do not have complex multiplication.

Modular form 197106.2.a.by

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 3 q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + 3 q^{10} + 3 q^{11} - q^{12} - q^{13} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + q^{18} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.