Properties

Label 432450.cl
Number of curves $2$
Conductor $432450$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cl1")
 
E.isogeny_class()
 

Elliptic curves in class 432450.cl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
432450.cl1 432450cl2 \([1, -1, 0, -112982067, -462296770659]\) \(-15777367606441/3574920\) \(-36139658540867173125000\) \([]\) \(66355200\) \(3.3206\)  
432450.cl2 432450cl1 \([1, -1, 0, 536058, -2207810034]\) \(1685159/209250\) \(-2115354623229738281250\) \([]\) \(22118400\) \(2.7713\) \(\Gamma_0(N)\)-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 432450.cl1.

Rank

sage: E.rank()
 

The elliptic curves in class 432450.cl have rank \(1\).

Complex multiplication

The elliptic curves in class 432450.cl do not have complex multiplication.

Modular form 432450.2.a.cl

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{7} - q^{8} - 3 q^{11} + 2 q^{13} - q^{14} + q^{16} + 5 q^{19} + O(q^{20})\) 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.