Properties

Label 678.e
Number of curves $2$
Conductor $678$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 678.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
678.e1 678d2 \([1, 0, 0, -7121, -2567403]\) \(-39934705050538129/2823126576537804\) \(-2823126576537804\) \([]\) \(3528\) \(1.6444\)  
678.e2 678d1 \([1, 0, 0, -1661, 26097]\) \(-506814405937489/4048994304\) \(-4048994304\) \([7]\) \(504\) \(0.67149\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 678.e have rank \(0\).

Complex multiplication

The elliptic curves in class 678.e do not have complex multiplication.

Modular form 678.2.a.e

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.