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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("id1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 432450.2.a.id

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 4 q^{7} + q^{8} - 3 q^{11} + 5 q^{13} + 4 q^{14} + q^{16} + 3 q^{17} - 7 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 432450.id

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
432450.id1 432450id1 \([1, -1, 1, -1157705, -1027383703]\) \(-458314011/953312\) \(-356934899169058500000\) \([]\) \(24883200\) \(2.6328\) \(\Gamma_0(N)\)-optimal*
432450.id2 432450id2 \([1, -1, 1, 10013920, 22202148547]\) \(406869021/1015808\) \(-277263761175968256000000\) \([]\) \(74649600\) \(3.1821\) \(\Gamma_0(N)\)-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 432450.id1.