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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 31680o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31680.n1 | 31680o1 | \([0, 0, 0, -4728, -124792]\) | \(15657723904/49005\) | \(36582036480\) | \([2]\) | \(32768\) | \(0.89428\) | \(\Gamma_0(N)\)-optimal |
31680.n2 | 31680o2 | \([0, 0, 0, -2748, -230128]\) | \(-192143824/1804275\) | \(-21550145126400\) | \([2]\) | \(65536\) | \(1.2409\) |
Rank
sage: E.rank()
The elliptic curves in class 31680o have rank \(0\).
Complex multiplication
The elliptic curves in class 31680o do not have complex multiplication.Modular form 31680.2.a.o
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 Cremona 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 Cremona labels.