Properties

Label 369600.pb
Number of curves $6$
Conductor $369600$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("pb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 369600.pb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
369600.pb1 369600pb6 [0, 1, 0, -4878720033, -131163302591937] [2] 94371840  
369600.pb2 369600pb4 [0, 1, 0, -304920033, -2049502391937] [2, 2] 47185920  
369600.pb3 369600pb5 [0, 1, 0, -303408033, -2070832175937] [2] 94371840  
369600.pb4 369600pb3 [0, 1, 0, -40712033, 52712856063] [2] 47185920  
369600.pb5 369600pb2 [0, 1, 0, -19152033, -31694543937] [2, 2] 23592960  
369600.pb6 369600pb1 [0, 1, 0, 55967, -1480359937] [2] 11796480 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 369600.pb have rank \(1\).

Complex multiplication

The elliptic curves in class 369600.pb do not have complex multiplication.

Modular form 369600.2.a.pb

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{7} + q^{9} + q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.