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Properties

Label 3969.d
Number of curves $2$
Conductor $3969$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3969.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3969.d1 3969a2 \([1, -1, 0, -170235, -26992176]\) \(1168429123449\) \(466948881\) \([]\) \(8820\) \(1.3978\)  
3969.d2 3969a1 \([1, -1, 0, -450, 3639]\) \(21609\) \(466948881\) \([]\) \(1260\) \(0.42486\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3969.d have rank \(1\).

Complex multiplication

The elliptic curves in class 3969.d do not have complex multiplication.

Modular form 3969.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} + 5 q^{11} - 5 q^{13} - q^{16} + 3 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.