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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 126852.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
126852.o1 | 126852m2 | \([0, 1, 0, -14349972, 20853963492]\) | \(1438357277593168/5107410363\) | \(1160408447370251827968\) | \([2]\) | \(8847360\) | \(2.9038\) | |
126852.o2 | 126852m1 | \([0, 1, 0, -497157, 617771340]\) | \(-957007003648/11062858059\) | \(-157093235995888242864\) | \([2]\) | \(4423680\) | \(2.5572\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 126852.o have rank \(1\).
Complex multiplication
The elliptic curves in class 126852.o do not have complex multiplication.Modular form 126852.2.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.