Properties

Label 627.a
Number of curves $2$
Conductor $627$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 627.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
627.a1 627b2 \([0, 1, 1, -30063, -2016358]\) \(-3004935183806464000/2037123\) \(-2037123\) \([]\) \(540\) \(0.96017\)  
627.a2 627b1 \([0, 1, 1, -363, -2995]\) \(-5304438784000/497763387\) \(-497763387\) \([3]\) \(180\) \(0.41087\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 627.a have rank \(0\).

Complex multiplication

The elliptic curves in class 627.a do not have complex multiplication.

Modular form 627.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.