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Properties

Label 61710bp
Number of curves $2$
Conductor $61710$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bp1")
 
E.isogeny_class()
 

Elliptic curves in class 61710bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.bt2 61710bp1 \([1, 1, 1, -2757411, 9659431233]\) \(-1308796492121439049/22000592486400000\) \(-38975391625799270400000\) \([2]\) \(5990400\) \(3.0160\) \(\Gamma_0(N)\)-optimal
61710.bt1 61710bp2 \([1, 1, 1, -87012131, 311190223169]\) \(41125104693338423360329/179205840000000000\) \(317474077116240000000000\) \([2]\) \(11980800\) \(3.3626\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 61710.2.a.bp

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.