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Properties

Label 61710be
Number of curves $2$
Conductor $61710$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("be1")
 
E.isogeny_class()
 

Elliptic curves in class 61710be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.s1 61710be1 \([1, 0, 1, -34004, -2888998]\) \(-2454365649169/610929000\) \(-1082297990169000\) \([]\) \(570240\) \(1.6028\) \(\Gamma_0(N)\)-optimal
61710.s2 61710be2 \([1, 0, 1, 245506, 20142626]\) \(923754305147471/633316406250\) \(-1121958645972656250\) \([]\) \(1710720\) \(2.1521\)  

Rank

sage: E.rank()
 

The elliptic curves in class 61710be have rank \(0\).

Complex multiplication

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

Modular form 61710.2.a.be

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.