Properties

Label 136045.s
Number of curves $2$
Conductor $136045$
CM no
Rank $0$
Graph

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

Elliptic curves in class 136045.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
136045.s1 136045s1 \([1, -1, 0, -27494, -1747225]\) \(476196576129/197225\) \(951967405025\) \([2]\) \(311040\) \(1.2605\) \(\Gamma_0(N)\)-optimal
136045.s2 136045s2 \([1, -1, 0, -23269, -2305770]\) \(-288673724529/311181605\) \(-1502014171648445\) \([2]\) \(622080\) \(1.6070\)  

Rank

sage: E.rank()
 

The elliptic curves in class 136045.s have rank \(0\).

Complex multiplication

The elliptic curves in class 136045.s do not have complex multiplication.

Modular form 136045.2.a.s

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