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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 72450cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.cs1 | 72450cm1 | \([1, -1, 1, -1108485380, 5939475630247]\) | \(489781415227546051766883/233890092903563264000\) | \(71932167165950558208000000000\) | \([2]\) | \(74317824\) | \(4.2310\) | \(\Gamma_0(N)\)-optimal |
| 72450.cs2 | 72450cm2 | \([1, -1, 1, 3978746620, 45172208814247]\) | \(22649115256119592694355357/15973509811739648000000\) | \(-4912603025382367056000000000000\) | \([2]\) | \(148635648\) | \(4.5776\) |
Rank
sage: E.rank()
The elliptic curves in class 72450cm have rank \(1\).
Complex multiplication
The elliptic curves in class 72450cm do not have complex multiplication.Modular form 72450.2.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 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.
