Properties

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

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

Elliptic curves in class 3450d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3450.d5 3450d1 \([1, 1, 0, -10500, 450000]\) \(-8194759433281/965779200\) \(-15090300000000\) \([2]\) \(9216\) \(1.2646\) \(\Gamma_0(N)\)-optimal
3450.d4 3450d2 \([1, 1, 0, -172500, 27504000]\) \(36330796409313601/428490000\) \(6695156250000\) \([2, 2]\) \(18432\) \(1.6112\)  
3450.d3 3450d3 \([1, 1, 0, -177000, 25987500]\) \(39248884582600321/3935264062500\) \(61488500976562500\) \([2, 2]\) \(36864\) \(1.9577\)  
3450.d1 3450d4 \([1, 1, 0, -2760000, 1763716500]\) \(148809678420065817601/20700\) \(323437500\) \([2]\) \(36864\) \(1.9577\)  
3450.d2 3450d5 \([1, 1, 0, -645750, -171356250]\) \(1905890658841300321/293666194803750\) \(4588534293808593750\) \([2]\) \(73728\) \(2.3043\)  
3450.d6 3450d6 \([1, 1, 0, 219750, 126365250]\) \(75108181893694559/484313964843750\) \(-7567405700683593750\) \([2]\) \(73728\) \(2.3043\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 3450d do not have complex multiplication.

Modular form 3450.2.a.d

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