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Properties

Label 9867.d
Number of curves $2$
Conductor $9867$
CM no
Rank $0$
Graph

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

Elliptic curves in class 9867.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9867.d1 9867e2 \([1, 1, 1, -8307, -292986]\) \(63395672188101553/475137974883\) \(475137974883\) \([2]\) \(14208\) \(1.0708\)  
9867.d2 9867e1 \([1, 1, 1, -8292, -294084]\) \(63052870949070913/3581721\) \(3581721\) \([2]\) \(7104\) \(0.72418\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 9867.d have rank \(0\).

Complex multiplication

The elliptic curves in class 9867.d do not have complex multiplication.

Modular form 9867.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + 2 q^{5} + q^{6} - 2 q^{7} + 3 q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + q^{13} + 2 q^{14} - 2 q^{15} - q^{16} + 2 q^{17} - q^{18} + 6 q^{19} + O(q^{20})\) 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.