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Properties

Label 413270.ci
Number of curves $4$
Conductor $413270$
CM no
Rank $2$
Graph

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Copy content sage:E = EllipticCurve("ci1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 413270.ci have rank \(2\).

Complex multiplication

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

Modular form 413270.2.a.ci

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 413270.ci

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
413270.ci1 413270ci3 \([1, -1, 1, -4267150513, 107290031901057]\) \(355995140004443961140387841/2768480\) \(66824377025120\) \([2]\) \(137625600\) \(3.6748\) \(\Gamma_0(N)\)-optimal*
413270.ci2 413270ci4 \([1, -1, 1, -267298033, 1668520212481]\) \(87501897507774086005761/815991377947460000\) \(19696048188611894124740000\) \([2]\) \(137625600\) \(3.6748\)  
413270.ci3 413270ci2 \([1, -1, 1, -266696913, 1676456679617]\) \(86912881496074271306241/7664481510400\) \(185001951306504217600\) \([2, 2]\) \(68812800\) \(3.3282\) \(\Gamma_0(N)\)-optimal*
413270.ci4 413270ci1 \([1, -1, 1, -16630993, 26321686721]\) \(-21075830718885163521/199306463150080\) \(-4810773506431013355520\) \([2]\) \(34406400\) \(2.9817\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 413270.ci1.