Properties

Label 83391.i
Number of curves $2$
Conductor $83391$
CM no
Rank $1$
Graph

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

Elliptic curves in class 83391.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
83391.i1 83391g2 \([0, -1, 1, -1001173, 387169956]\) \(-2359010787328000/8925676683\) \(-419916323072892723\) \([]\) \(1088640\) \(2.2409\)  
83391.i2 83391g1 \([0, -1, 1, 27677, 2771019]\) \(49836032000/99819027\) \(-4696074065777787\) \([]\) \(362880\) \(1.6916\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 83391.i have rank \(1\).

Complex multiplication

The elliptic curves in class 83391.i do not have complex multiplication.

Modular form 83391.2.a.i

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.