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Properties

Label 412269b
Number of curves $6$
Conductor $412269$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 412269b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412269.b5 412269b1 \([1, 1, 1, -23084, -1946068]\) \(-1532808577/938223\) \(-832676366098863\) \([2]\) \(1966080\) \(1.5649\) \(\Gamma_0(N)\)-optimal*
412269.b4 412269b2 \([1, 1, 1, -412289, -102049594]\) \(8732907467857/1656369\) \(1470033584594289\) \([2, 2]\) \(3932160\) \(1.9115\) \(\Gamma_0(N)\)-optimal*
412269.b3 412269b3 \([1, 1, 1, -455534, -79389214]\) \(11779205551777/3763454409\) \(3340079641263179529\) \([2, 2]\) \(7864320\) \(2.2581\) \(\Gamma_0(N)\)-optimal*
412269.b1 412269b4 \([1, 1, 1, -6596324, -6523551538]\) \(35765103905346817/1287\) \(1142217237447\) \([2]\) \(7864320\) \(2.2581\)  
412269.b2 412269b5 \([1, 1, 1, -2891669, 1831515080]\) \(3013001140430737/108679952667\) \(96453858042868267227\) \([2]\) \(15728640\) \(2.6047\) \(\Gamma_0(N)\)-optimal*
412269.b6 412269b6 \([1, 1, 1, 1288681, -539164288]\) \(266679605718863/296110251723\) \(-262798938385999092363\) \([2]\) \(15728640\) \(2.6047\)  
*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 412269b1.

Rank

sage: E.rank()
 

The elliptic curves in class 412269b have rank \(1\).

Complex multiplication

The elliptic curves in class 412269b do not have complex multiplication.

Modular form 412269.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 3 q^{8} + q^{9} + 2 q^{10} + q^{11} + q^{12} - q^{13} + 2 q^{15} - q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.