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Properties

Label 61710.bz
Number of curves $4$
Conductor $61710$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("bz1")
 
E.isogeny_class()
 

Elliptic curves in class 61710.bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.bz1 61710ca4 \([1, 1, 1, -68721650, -219303304945]\) \(20260414982443110947641/720358602480\) \(1276159206168071280\) \([2]\) \(4423680\) \(2.9700\)  
61710.bz2 61710ca2 \([1, 1, 1, -4301250, -3417660465]\) \(4967657717692586041/29490113030400\) \(52243534130248454400\) \([2, 2]\) \(2211840\) \(2.6234\)  
61710.bz3 61710ca3 \([1, 1, 1, -1832850, -7309833585]\) \(-384369029857072441/12804787777021680\) \(-22684462639048304442480\) \([2]\) \(4423680\) \(2.9700\)  
61710.bz4 61710ca1 \([1, 1, 1, -429250, 17577935]\) \(4937402992298041/2780405760000\) \(4925658408591360000\) \([4]\) \(1105920\) \(2.2768\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 61710.2.a.bz

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.