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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 67830.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67830.o1 | 67830m2 | \([1, 0, 1, -220119, -39767558]\) | \(1179487844109684177769/13959414000000\) | \(13959414000000\) | \([2]\) | \(559104\) | \(1.6722\) | |
67830.o2 | 67830m1 | \([1, 0, 1, -13399, -656134]\) | \(-266007743865635689/31408871424000\) | \(-31408871424000\) | \([2]\) | \(279552\) | \(1.3256\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 67830.o have rank \(1\).
Complex multiplication
The elliptic curves in class 67830.o do not have complex multiplication.Modular form 67830.2.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 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.