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Properties

Label 491970d
Number of curves $2$
Conductor $491970$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 491970d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
491970.d2 491970d1 \([1, 1, 0, 1183627, -2193261267]\) \(1238798620042199/14760960000000\) \(-2185151836093440000000\) \([2]\) \(34062336\) \(2.7756\) \(\Gamma_0(N)\)-optimal*
491970.d1 491970d2 \([1, 1, 0, -19807093, -31660034003]\) \(5805223604235668521/435937500000000\) \(64534395360937500000000\) \([2]\) \(68124672\) \(3.1221\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 491970d1.

Rank

sage: E.rank()
 

The elliptic curves in class 491970d have rank \(1\).

Complex multiplication

The elliptic curves in class 491970d do not have complex multiplication.

Modular form 491970.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} - 4 q^{13} + 2 q^{14} + q^{15} + q^{16} + 6 q^{17} - q^{18} + O(q^{20})\) 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.