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Properties

Label 494190.q
Number of curves $4$
Conductor $494190$
CM no
Rank $1$
Graph

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Copy content sage:E = EllipticCurve("q1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 494190.q have rank \(1\).

Complex multiplication

The elliptic curves in class 494190.q do not have complex multiplication.

Modular form 494190.2.a.q

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 494190.q

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
494190.q1 494190q4 \([1, -1, 0, -1204863885, -10278634232075]\) \(10993009831928446009969/3767761230468750000\) \(66298610976778015136718750000\) \([2]\) \(637009920\) \(4.2323\) \(\Gamma_0(N)\)-optimal*
494190.q2 494190q2 \([1, -1, 0, -1079391645, -13649227225979]\) \(7903870428425797297009/886464000000\) \(15598475669225664000000\) \([2]\) \(212336640\) \(3.6830\) \(\Gamma_0(N)\)-optimal*
494190.q3 494190q1 \([1, -1, 0, -67290525, -214394538875]\) \(-1914980734749238129/20440940544000\) \(-359684672735353503744000\) \([2]\) \(106168320\) \(3.3364\) \(\Gamma_0(N)\)-optimal*
494190.q4 494190q3 \([1, -1, 0, 222356835, -1116162653819]\) \(69096190760262356111/70568821500000000\) \(-1241749292891396521500000000\) \([2]\) \(318504960\) \(3.8857\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 494190.q1.