Properties

Label 405042.y
Number of curves $2$
Conductor $405042$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("y1")
 
E.isogeny_class()
 

Elliptic curves in class 405042.y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405042.y1 405042y2 \([1, 1, 0, -65709, -6510285]\) \(666940371553/37026\) \(1741920789906\) \([2]\) \(1368576\) \(1.4142\)  
405042.y2 405042y1 \([1, 1, 0, -4339, -90983]\) \(192100033/38148\) \(1794706268388\) \([2]\) \(684288\) \(1.0676\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 405042.y1.

Rank

sage: E.rank()
 

The elliptic curves in class 405042.y have rank \(1\).

Complex multiplication

The elliptic curves in class 405042.y do not have complex multiplication.

Modular form 405042.2.a.y

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