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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 12342.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12342.s1 | 12342r1 | \([1, 0, 1, -23719, 1403114]\) | \(832972004929/610368\) | \(1081304144448\) | \([2]\) | \(46080\) | \(1.2433\) | \(\Gamma_0(N)\)-optimal |
12342.s2 | 12342r2 | \([1, 0, 1, -18879, 1993594]\) | \(-420021471169/727634952\) | \(-1289049703200072\) | \([2]\) | \(92160\) | \(1.5899\) |
Rank
sage: E.rank()
The elliptic curves in class 12342.s have rank \(0\).
Complex multiplication
The elliptic curves in class 12342.s do not have complex multiplication.Modular form 12342.2.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.