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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)
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.