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Properties

Label 432450gm
Number of curves $2$
Conductor $432450$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 432450gm have rank \(0\).

Complex multiplication

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

Modular form 432450.2.a.gm

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} + 4 q^{11} - 4 q^{13} + q^{16} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 432450gm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
432450.gm2 432450gm1 \([1, -1, 1, -92046605, 351850393397]\) \(-254164210474783519/10497600000000\) \(-3562235736975000000000000\) \([2]\) \(94371840\) \(3.4782\) \(\Gamma_0(N)\)-optimal*
432450.gm1 432450gm2 \([1, -1, 1, -1487046605, 22072000393397]\) \(1071679233972039583519/1721868840000\) \(584295716757324375000000\) \([2]\) \(188743680\) \(3.8248\) \(\Gamma_0(N)\)-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 432450gm1.