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Properties

Label 418275.cg
Number of curves $6$
Conductor $418275$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cg1")
 
E.isogeny_class()
 

Elliptic curves in class 418275.cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418275.cg1 418275cg4 \([1, -1, 0, -261004392, -1622939671859]\) \(35765103905346817/1287\) \(70759737818859375\) \([2]\) \(44040192\) \(3.1776\)  
418275.cg2 418275cg5 \([1, -1, 0, -114418017, 456186144766]\) \(3013001140430737/108679952667\) \(5975264146762211884171875\) \([2]\) \(88080384\) \(3.5242\) \(\Gamma_0(N)\)-optimal*
418275.cg3 418275cg3 \([1, -1, 0, -18024642, -19707947609]\) \(11779205551777/3763454409\) \(206916120648073316390625\) \([2, 2]\) \(44040192\) \(3.1776\) \(\Gamma_0(N)\)-optimal*
418275.cg4 418275cg2 \([1, -1, 0, -16313517, -25352948984]\) \(8732907467857/1656369\) \(91067782572872015625\) \([2, 2]\) \(22020096\) \(2.8310\) \(\Gamma_0(N)\)-optimal*
418275.cg5 418275cg1 \([1, -1, 0, -913392, -481747109]\) \(-1532808577/938223\) \(-51583848869948484375\) \([2]\) \(11010048\) \(2.4844\) \(\Gamma_0(N)\)-optimal*
418275.cg6 418275cg6 \([1, -1, 0, 50990733, -134342485484]\) \(266679605718863/296110251723\) \(-16280251575288214846921875\) \([2]\) \(88080384\) \(3.5242\)  
*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 418275.cg1.

Rank

sage: E.rank()
 

The elliptic curves in class 418275.cg have rank \(0\).

Complex multiplication

The elliptic curves in class 418275.cg do not have complex multiplication.

Modular form 418275.2.a.cg

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

Isogeny matrix

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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.