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Properties

Label 413270cx
Number of curves $4$
Conductor $413270$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cx1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 413270cx have rank \(0\).

Complex multiplication

The elliptic curves in class 413270cx do not have complex multiplication.

Modular form 413270.2.a.cx

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 q^{6} + 4 q^{7} + q^{8} + q^{9} + q^{10} - q^{11} + 2 q^{12} + q^{13} + 4 q^{14} + 2 q^{15} + q^{16} + q^{18} + 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 413270cx

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
413270.cx4 413270cx1 \([1, 1, 1, 30050, -30053]\) \(124326214271/71980480\) \(-1737433802653120\) \([2]\) \(3538944\) \(1.6137\) \(\Gamma_0(N)\)-optimal*
413270.cx3 413270cx2 \([1, 1, 1, -120230, -390725]\) \(7962857630209/4606058600\) \(111179057275543400\) \([2]\) \(7077888\) \(1.9603\) \(\Gamma_0(N)\)-optimal*
413270.cx2 413270cx3 \([1, 1, 1, -415010, -110048885]\) \(-327495950129089/26547449500\) \(-640790894080265500\) \([2]\) \(10616832\) \(2.1630\)  
413270.cx1 413270cx4 \([1, 1, 1, -6764340, -6774305653]\) \(1418098748958579169/8307406250\) \(200520591570406250\) \([2]\) \(21233664\) \(2.5096\)  
*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 413270cx1.