Properties

Label 3366.d
Number of curves $4$
Conductor $3366$
CM no
Rank $0$
Graph

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

Elliptic curves in class 3366.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3366.d1 3366c3 \([1, -1, 0, -17433, 825475]\) \(803760366578833/65593817586\) \(47817893020194\) \([2]\) \(12288\) \(1.3676\)  
3366.d2 3366c2 \([1, -1, 0, -3663, -69575]\) \(7457162887153/1370924676\) \(999404088804\) \([2, 2]\) \(6144\) \(1.0210\)  
3366.d3 3366c1 \([1, -1, 0, -3483, -78251]\) \(6411014266033/296208\) \(215935632\) \([2]\) \(3072\) \(0.67442\) \(\Gamma_0(N)\)-optimal
3366.d4 3366c4 \([1, -1, 0, 7227, -411521]\) \(57258048889007/132611470002\) \(-96673761631458\) \([2]\) \(12288\) \(1.3676\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3366.d have rank \(0\).

Complex multiplication

The elliptic curves in class 3366.d do not have complex multiplication.

Modular form 3366.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} + 4 q^{7} - q^{8} + 2 q^{10} - q^{11} - 2 q^{13} - 4 q^{14} + q^{16} - q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.