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SageMath
E = EllipticCurve("pb1")
E.isogeny_class()
Elliptic curves in class 381150pb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 381150.pb7 | 381150pb1 | \([1, -1, 1, -31921880, 31569468747]\) | \(178272935636041/81841914000\) | \(1651503288322697906250000\) | \([2]\) | \(53084160\) | \(3.3418\) | \(\Gamma_0(N)\)-optimal |
| 381150.pb5 | 381150pb2 | \([1, -1, 1, -428862380, 3416678052747]\) | \(432288716775559561/270140062500\) | \(5451206841600125976562500\) | \([2, 2]\) | \(106168320\) | \(3.6884\) | |
| 381150.pb4 | 381150pb3 | \([1, -1, 1, -1299926255, -18038264703753]\) | \(12038605770121350841/757333463040\) | \(15282373583875083840000000\) | \([2]\) | \(159252480\) | \(3.8911\) | |
| 381150.pb2 | 381150pb4 | \([1, -1, 1, -6860768630, 218731171677747]\) | \(1769857772964702379561/691787250\) | \(13959704292774925781250\) | \([2]\) | \(212336640\) | \(4.0350\) | |
| 381150.pb6 | 381150pb5 | \([1, -1, 1, -348004130, 4744370517747]\) | \(-230979395175477481/348191894531250\) | \(-7026229357070869445800781250\) | \([2]\) | \(212336640\) | \(4.0350\) | |
| 381150.pb3 | 381150pb6 | \([1, -1, 1, -1378334255, -15739498959753]\) | \(14351050585434661561/3001282273281600\) | \(60563436279229545579225000000\) | \([2, 2]\) | \(318504960\) | \(4.2377\) | |
| 381150.pb1 | 381150pb7 | \([1, -1, 1, -6981239255, 210651480470247]\) | \(1864737106103260904761/129177711985836360\) | \(2606701208412735009618013125000\) | \([2]\) | \(637009920\) | \(4.5843\) | |
| 381150.pb8 | 381150pb8 | \([1, -1, 1, 2970042745, -95010411669753]\) | \(143584693754978072519/276341298967965000\) | \(-5576342752015305540829453125000\) | \([2]\) | \(637009920\) | \(4.5843\) |
Rank
sage: E.rank()
The elliptic curves in class 381150pb have rank \(1\).
Complex multiplication
The elliptic curves in class 381150pb do not have complex multiplication.Modular form 381150.2.a.pb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
