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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 67158.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67158.s1 | 67158m2 | \([1, -1, 0, -80987571, -280507403403]\) | \(80584461459025151375298097/333693103021824\) | \(243262272102909696\) | \([2]\) | \(4423680\) | \(2.9682\) | |
67158.s2 | 67158m1 | \([1, -1, 0, -5059251, -4386474891]\) | \(-19645130164017251655217/40036908053495808\) | \(-29186905970998444032\) | \([2]\) | \(2211840\) | \(2.6217\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 67158.s have rank \(1\).
Complex multiplication
The elliptic curves in class 67158.s do not have complex multiplication.Modular form 67158.2.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.