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Properties

Label 1386.e
Number of curves $2$
Conductor $1386$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 1386.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1386.e1 1386e2 \([1, -1, 0, -67005, -6657467]\) \(45637459887836881/13417633152\) \(9781454567808\) \([2]\) \(10752\) \(1.4716\)  
1386.e2 1386e1 \([1, -1, 0, -3645, -131387]\) \(-7347774183121/6119866368\) \(-4461382582272\) \([2]\) \(5376\) \(1.1251\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1386.e have rank \(0\).

Complex multiplication

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

Modular form 1386.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.