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Properties

Label 67270.s
Number of curves $2$
Conductor $67270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 67270.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67270.s1 67270l2 \([1, 1, 0, -142345742, 653620120244]\) \(12064294055330551/560000\) \(14806188409975760000\) \([2]\) \(8888320\) \(3.1576\)  
67270.s2 67270l1 \([1, 1, 0, -8882062, 10245104436]\) \(-2930960362231/20070400\) \(-530653792613531238400\) \([2]\) \(4444160\) \(2.8110\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 67270.s have rank \(1\).

Complex multiplication

The elliptic curves in class 67270.s do not have complex multiplication.

Modular form 67270.2.a.s

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