Properties

Label 3450d
Number of curves $6$
Conductor $3450$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("3450.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3450d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3450.d5 3450d1 [1, 1, 0, -10500, 450000] [2] 9216 \(\Gamma_0(N)\)-optimal
3450.d4 3450d2 [1, 1, 0, -172500, 27504000] [2, 2] 18432  
3450.d3 3450d3 [1, 1, 0, -177000, 25987500] [2, 2] 36864  
3450.d1 3450d4 [1, 1, 0, -2760000, 1763716500] [2] 36864  
3450.d2 3450d5 [1, 1, 0, -645750, -171356250] [2] 73728  
3450.d6 3450d6 [1, 1, 0, 219750, 126365250] [2] 73728  

Rank

sage: E.rank()
 

The elliptic curves in class 3450d have rank \(0\).

Modular form 3450.2.a.d

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

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.