Properties

Label 1386.d
Number of curves $2$
Conductor $1386$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1386.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1386.d1 1386a1 \([1, -1, 0, -231, -451]\) \(69426531/34496\) \(678984768\) \([2]\) \(576\) \(0.38764\) \(\Gamma_0(N)\)-optimal
1386.d2 1386a2 \([1, -1, 0, 849, -4123]\) \(3436115229/2324168\) \(-45746598744\) \([2]\) \(1152\) \(0.73421\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1386.d have rank \(1\).

Complex multiplication

The elliptic curves in class 1386.d do not have complex multiplication.

Modular form 1386.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 2q^{5} - q^{7} - q^{8} - 2q^{10} - q^{11} - 4q^{13} + q^{14} + q^{16} + 2q^{17} - 6q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.