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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 81312.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
81312.o1 | 81312bd2 | \([0, -1, 0, -4033, -91727]\) | \(1000000/63\) | \(457147772928\) | \([2]\) | \(89600\) | \(0.98840\) | |
81312.o2 | 81312bd1 | \([0, -1, 0, 202, -6180]\) | \(8000/147\) | \(-16666845888\) | \([2]\) | \(44800\) | \(0.64183\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 81312.o have rank \(0\).
Complex multiplication
The elliptic curves in class 81312.o do not have complex multiplication.Modular form 81312.2.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.