↑

Properties

Label 61710.by
Number of curves $4$
Conductor $61710$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 61710.by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.by1 61710bz4 \([1, 1, 1, -45605810, -115049990263]\) \(5921450764096952391481/200074809015963750\) \(354444728735129756913750\) \([2]\) \(8847360\) \(3.2911\)  
61710.by2 61710bz2 \([1, 1, 1, -7037060, 4698264737]\) \(21754112339458491481/7199734626562500\) \(12754769074767689062500\) \([2, 2]\) \(4423680\) \(2.9445\)  
61710.by3 61710bz1 \([1, 1, 1, -6337680, 6137309025]\) \(15891267085572193561/3334993530000\) \(5908144473000330000\) \([4]\) \(2211840\) \(2.5979\) \(\Gamma_0(N)\)-optimal
61710.by4 61710bz3 \([1, 1, 1, 20341610, 32361672905]\) \(525440531549759128199/559322204589843750\) \(-990873404085388183593750\) \([2]\) \(8847360\) \(3.2911\)  

Rank

sage: E.rank()
 

The elliptic curves in class 61710.by have rank \(1\).

Complex multiplication

The elliptic curves in class 61710.by do not have complex multiplication.

Modular form 61710.2.a.by

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