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Properties

Label 411840.dr
Number of curves $4$
Conductor $411840$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 411840.dr have rank \(0\).

Complex multiplication

The elliptic curves in class 411840.dr do not have complex multiplication.

Modular form 411840.2.a.dr

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} + q^{11} + q^{13} - 6 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 411840.dr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.dr1 411840dr4 \([0, 0, 0, -209351282988, -36869016557889488]\) \(5309860874757074224246393258249/4502770931800627200\) \(860492925313392896586547200\) \([2]\) \(1038090240\) \(4.9032\)  
411840.dr2 411840dr2 \([0, 0, 0, -13087346988, -575811007195088]\) \(1297212465095901089487274249/1193746061037404160000\) \(228128424852525582290780160000\) \([2, 2]\) \(519045120\) \(4.5567\)  
411840.dr3 411840dr3 \([0, 0, 0, -10096939308, -845936925255632]\) \(-595697118196750093952139529/1272946549598037600000000\) \(-243263873917116589119897600000000\) \([2]\) \(1038090240\) \(4.9032\)  
411840.dr4 411840dr1 \([0, 0, 0, -1007751468, -4508952995792]\) \(592265697637387401314569/296787655248366796800\) \(56717004158024914008067276800\) \([2]\) \(259522560\) \(4.2101\) \(\Gamma_0(N)\)-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 411840.dr1.