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SageMath
E = EllipticCurve("pc1")
E.isogeny_class()
Elliptic curves in class 187200.pc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.pc1 | 187200kk2 | \([0, 0, 0, -8460, -291600]\) | \(5606442/169\) | \(2018525184000\) | \([2]\) | \(327680\) | \(1.1375\) | |
187200.pc2 | 187200kk1 | \([0, 0, 0, -1260, 10800]\) | \(37044/13\) | \(77635584000\) | \([2]\) | \(163840\) | \(0.79096\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.pc have rank \(1\).
Complex multiplication
The elliptic curves in class 187200.pc do not have complex multiplication.Modular form 187200.2.a.pc
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.