Properties

Label 3330.d
Number of curves $4$
Conductor $3330$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3330.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3330.d1 3330g3 \([1, -1, 0, -47475, 3993381]\) \(16232905099479601/4052240\) \(2954082960\) \([6]\) \(6912\) \(1.1928\)  
3330.d2 3330g4 \([1, -1, 0, -47295, 4025025]\) \(-16048965315233521/256572640900\) \(-187041455216100\) \([6]\) \(13824\) \(1.5394\)  
3330.d3 3330g1 \([1, -1, 0, -675, 3861]\) \(46694890801/18944000\) \(13810176000\) \([2]\) \(2304\) \(0.64351\) \(\Gamma_0(N)\)-optimal
3330.d4 3330g2 \([1, -1, 0, 2205, 26325]\) \(1625964918479/1369000000\) \(-998001000000\) \([2]\) \(4608\) \(0.99008\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3330.d have rank \(1\).

Complex multiplication

The elliptic curves in class 3330.d do not have complex multiplication.

Modular form 3330.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} + 2q^{7} - q^{8} + q^{10} + 2q^{13} - 2q^{14} + q^{16} - 6q^{17} + 2q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.