Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("412269.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 412269b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
412269.b5 412269b1 [1, 1, 1, -23084, -1946068] [2] 1966080 \(\Gamma_0(N)\)-optimal*
412269.b4 412269b2 [1, 1, 1, -412289, -102049594] [2, 2] 3932160 \(\Gamma_0(N)\)-optimal*
412269.b3 412269b3 [1, 1, 1, -455534, -79389214] [2, 2] 7864320 \(\Gamma_0(N)\)-optimal*
412269.b1 412269b4 [1, 1, 1, -6596324, -6523551538] [2] 7864320  
412269.b2 412269b5 [1, 1, 1, -2891669, 1831515080] [2] 15728640 \(\Gamma_0(N)\)-optimal*
412269.b6 412269b6 [1, 1, 1, 1288681, -539164288] [2] 15728640  
*optimality has not been proved 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 \(1\).

Modular form 412269.2.a.b

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

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.