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Properties

Label 277350.cp
Number of curves $2$
Conductor $277350$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cp1")
 
E.isogeny_class()
 

Elliptic curves in class 277350.cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
277350.cp1 277350cp2 \([1, 1, 1, -61757563, -186825967969]\) \(263732349218689/4160250\) \(410913291009410156250\) \([2]\) \(25546752\) \(3.0893\)  
277350.cp2 277350cp1 \([1, 1, 1, -3976313, -2734905469]\) \(70393838689/8062500\) \(796343587227539062500\) \([2]\) \(12773376\) \(2.7428\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 277350.cp have rank \(1\).

Complex multiplication

The elliptic curves in class 277350.cp do not have complex multiplication.

Modular form 277350.2.a.cp

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