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SageMath
sage: E = EllipticCurve("p1")
sage: E.isogeny_class()
Elliptic curves in class 72450.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.p1 | 72450u3 | \([1, -1, 0, -649692, 66451216]\) | \(2662558086295801/1374177967680\) | \(15652745913105000000\) | \([2]\) | \(1658880\) | \(2.3752\) | |
| 72450.p2 | 72450u1 | \([1, -1, 0, -362817, -84023159]\) | \(463702796512201/15214500\) | \(173302664062500\) | \([2]\) | \(552960\) | \(1.8259\) | \(\Gamma_0(N)\)-optimal |
| 72450.p3 | 72450u2 | \([1, -1, 0, -347067, -91661909]\) | \(-405897921250921/84358968750\) | \(-960901378417968750\) | \([2]\) | \(1105920\) | \(2.1724\) | |
| 72450.p4 | 72450u4 | \([1, -1, 0, 2437308, 514066216]\) | \(140574743422291079/91397357868600\) | \(-1041073029472021875000\) | \([2]\) | \(3317760\) | \(2.7218\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.p have rank \(0\).
Complex multiplication
The elliptic curves in class 72450.p do not have complex multiplication.Modular form 72450.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
