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Properties

Label 21866c
Number of curves $2$
Conductor $21866$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 21866c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21866.e2 21866c1 \([1, -1, 0, -2260, 63824]\) \(-2146689/1664\) \(-989786006144\) \([]\) \(50176\) \(1.0005\) \(\Gamma_0(N)\)-optimal
21866.e1 21866c2 \([1, -1, 0, -178870, -31902586]\) \(-1064019559329/125497034\) \(-74648562539529914\) \([]\) \(351232\) \(1.9734\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21866c have rank \(1\).

Complex multiplication

The elliptic curves in class 21866c do not have complex multiplication.

Modular form 21866.2.a.c

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