Properties

Label 247938.p
Number of curves $2$
Conductor $247938$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 247938.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
247938.p1 247938p2 \([1, 1, 1, -57564881, 168102454067]\) \(-23769846831649063249/3261823333284\) \(-2894880215061239818404\) \([]\) \(32598720\) \(3.1359\)  
247938.p2 247938p1 \([1, 1, 1, 152779, -51288613]\) \(444369620591/1540767744\) \(-1367437044366065664\) \([]\) \(4656960\) \(2.1630\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 247938.p have rank \(1\).

Complex multiplication

The elliptic curves in class 247938.p do not have complex multiplication.

Modular form 247938.2.a.p

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