Properties

Label 1386.k
Number of curves $4$
Conductor $1386$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1386.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1386.k1 1386l4 \([1, -1, 1, -2039, 35925]\) \(1285429208617/614922\) \(448278138\) \([2]\) \(1024\) \(0.61497\)  
1386.k2 1386l3 \([1, -1, 1, -1139, -14259]\) \(223980311017/4278582\) \(3119086278\) \([2]\) \(1024\) \(0.61497\)  
1386.k3 1386l2 \([1, -1, 1, -149, 393]\) \(498677257/213444\) \(155600676\) \([2, 2]\) \(512\) \(0.26839\)  
1386.k4 1386l1 \([1, -1, 1, 31, 33]\) \(4657463/3696\) \(-2694384\) \([2]\) \(256\) \(-0.078180\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1386.2.a.k

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.