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Properties

Label 459186by
Number of curves $3$
Conductor $459186$
CM no
Rank $0$
Graph

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

Elliptic curves in class 459186by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
459186.by3 459186by1 \([1, 1, 1, 11336, 4422209]\) \(270840023/14329224\) \(-8523356607032904\) \([]\) \(4953312\) \(1.7365\) \(\Gamma_0(N)\)-optimal
459186.by2 459186by2 \([1, 1, 1, -102199, -120466291]\) \(-198461344537/10417365504\) \(-6196491945160118784\) \([]\) \(14859936\) \(2.2858\)  
459186.by1 459186by3 \([1, 1, 1, -21913534, -39493369861]\) \(-1956469094246217097/36641439744\) \(-21795182874747469824\) \([]\) \(44579808\) \(2.8351\)  

Rank

sage: E.rank()
 

The elliptic curves in class 459186by have rank \(0\).

Complex multiplication

The elliptic curves in class 459186by do not have complex multiplication.

Modular form 459186.2.a.by

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 3 q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + 3 q^{10} - 3 q^{11} - q^{12} + q^{13} + q^{14} - 3 q^{15} + q^{16} + 3 q^{17} + q^{18} + 7 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.