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Properties

Label 61710b
Number of curves $2$
Conductor $61710$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 61710b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.f2 61710b1 \([1, 1, 0, -53363, 4712217]\) \(9486391169809/23842500\) \(42238443142500\) \([2]\) \(307200\) \(1.4916\) \(\Gamma_0(N)\)-optimal
61710.f1 61710b2 \([1, 1, 0, -73933, 717523]\) \(25228519578289/14463281250\) \(25622584994531250\) \([2]\) \(614400\) \(1.8382\)  

Rank

sage: E.rank()
 

The elliptic curves in class 61710b have rank \(2\).

Complex multiplication

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

Modular form 61710.2.a.b

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