Properties

Label 72450.dp
Number of curves $6$
Conductor $72450$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("72450.dp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 72450.dp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
72450.dp1 72450dn6 [1, -1, 1, -17805155, 28922059847] [2] 4194304  
72450.dp2 72450dn4 [1, -1, 1, -3985655, -3061573153] [2] 2097152  
72450.dp3 72450dn3 [1, -1, 1, -1141655, 427474847] [2, 2] 2097152  
72450.dp4 72450dn2 [1, -1, 1, -259655, -43513153] [2, 2] 1048576  
72450.dp5 72450dn1 [1, -1, 1, 28345, -3769153] [2] 524288 \(\Gamma_0(N)\)-optimal
72450.dp6 72450dn5 [1, -1, 1, 1409845, 2065537847] [2] 4194304  

Rank

sage: E.rank()
 

The elliptic curves in class 72450.dp have rank \(1\).

Modular form 72450.2.a.dp

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{7} + q^{8} + 4q^{11} + 2q^{13} - q^{14} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.