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Properties

Label 418950.hm
Number of curves $2$
Conductor $418950$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 418950.hm have rank \(0\).

Complex multiplication

The elliptic curves in class 418950.hm do not have complex multiplication.

Modular form 418950.2.a.hm

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} + 3 q^{11} + 6 q^{13} + q^{16} - 2 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 418950.hm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418950.hm1 418950hm2 \([1, -1, 0, -219127617, -1248667902959]\) \(-1389310279182025/267418692\) \(-223979139606774726562500\) \([]\) \(103680000\) \(3.4809\)  
418950.hm2 418950hm1 \([1, -1, 0, 2104443, 69434581]\) \(480705753733655/279172334592\) \(-598588205711748940800\) \([]\) \(20736000\) \(2.6762\) \(\Gamma_0(N)\)-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 418950.hm1.