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Properties

Label 476850.ds
Number of curves $2$
Conductor $476850$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("ds1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 476850.ds

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
476850.ds1 476850ds2 [1, 0, 1, -1315101, 580341598] [2] 7077888 \(\Gamma_0(N)\)-optimal*
476850.ds2 476850ds1 [1, 0, 1, -86851, 7977098] [2] 3538944 \(\Gamma_0(N)\)-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 476850.ds1.

Rank

sage: E.rank()
 

The elliptic curves in class 476850.ds have rank \(1\).

Complex multiplication

The elliptic curves in class 476850.ds do not have complex multiplication.

Modular form 476850.2.a.ds

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} + q^{4} - q^{6} - 2q^{7} - q^{8} + q^{9} + q^{11} + q^{12} - 4q^{13} + 2q^{14} + q^{16} - q^{18} + 2q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.