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Properties

Label 8670.c
Number of curves $2$
Conductor $8670$
CM no
Rank $1$
Graph

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

Elliptic curves in class 8670.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8670.c1 8670c2 \([1, 1, 0, -3353128, -2364721472]\) \(172735174415217961/39657600\) \(957238056374400\) \([2]\) \(193536\) \(2.2553\)  
8670.c2 8670c1 \([1, 1, 0, -208808, -37295808]\) \(-41713327443241/639221760\) \(-15429259338301440\) \([2]\) \(96768\) \(1.9087\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8670.c have rank \(1\).

Complex multiplication

The elliptic curves in class 8670.c do not have complex multiplication.

Modular form 8670.2.a.c

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