Properties

Label 414736.p
Number of curves $2$
Conductor $414736$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 414736.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.p1 414736p2 \([0, -1, 0, -3689422, -2726395741]\) \(406749952\) \(13654359055089424\) \([]\) \(6286896\) \(2.3357\)  
414736.p2 414736p1 \([0, -1, 0, -60482, -1061801]\) \(1792\) \(13654359055089424\) \([]\) \(2095632\) \(1.7864\) \(\Gamma_0(N)\)-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 414736.p1.

Rank

sage: E.rank()
 

The elliptic curves in class 414736.p have rank \(1\).

Complex multiplication

The elliptic curves in class 414736.p do not have complex multiplication.

Modular form 414736.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{3} - 3 q^{5} - 2 q^{9} - 3 q^{11} + 2 q^{13} + 3 q^{15} - 3 q^{17} - q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.