Properties

Label 6864.i
Number of curves $2$
Conductor $6864$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 6864.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6864.i1 6864a2 \([0, -1, 0, -40888, 2941984]\) \(7382814913718500/654774260283\) \(670488842529792\) \([2]\) \(21504\) \(1.5850\)  
6864.i2 6864a1 \([0, -1, 0, 2852, 212608]\) \(10017976862000/82759712607\) \(-21186486427392\) \([2]\) \(10752\) \(1.2384\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6864.i have rank \(1\).

Complex multiplication

The elliptic curves in class 6864.i do not have complex multiplication.

Modular form 6864.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{9} - q^{11} - q^{13} - 4 q^{17} + O(q^{20})\) 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.