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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 374850.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
374850.cm1 | 374850cm2 | \([1, -1, 0, -5716692, 3877132216]\) | \(15417797707369/4080067320\) | \(5467680428988526875000\) | \([2]\) | \(21233664\) | \(2.8797\) | |
374850.cm2 | 374850cm1 | \([1, -1, 0, 898308, 391027216]\) | \(59822347031/83966400\) | \(-112523006598975000000\) | \([2]\) | \(10616832\) | \(2.5331\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 374850.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 374850.cm do not have complex multiplication.Modular form 374850.2.a.cm
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.