Properties

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

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 476850.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
476850.x1 476850x2 \([1, 1, 0, -174060, -28023600]\) \(949659180759757/46464\) \(28534704000\) \([2]\) \(1720320\) \(1.4821\) \(\Gamma_0(N)\)-optimal*
476850.x2 476850x1 \([1, 1, 0, -10860, -442800]\) \(-230684754637/1622016\) \(-996120576000\) \([2]\) \(860160\) \(1.1355\) \(\Gamma_0(N)\)-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.x1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 476850.2.a.x

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