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SageMath
E = EllipticCurve("fm1")
E.isogeny_class()
Elliptic curves in class 381150fm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 381150.fm5 | 381150fm1 | \([1, -1, 0, -9061841817, 389040121783341]\) | \(-4078208988807294650401/880065599546327040\) | \(-17759008319263473496227840000000\) | \([2]\) | \(1061683200\) | \(4.7173\) | \(\Gamma_0(N)\)-optimal |
| 381150.fm4 | 381150fm2 | \([1, -1, 0, -151799249817, 22763557038007341]\) | \(19170300594578891358373921/671785075055001600\) | \(13556076663840554369433600000000\) | \([2, 2]\) | \(2123366400\) | \(5.0638\) | |
| 381150.fm1 | 381150fm3 | \([1, -1, 0, -2428767569817, 1456891560675127341]\) | \(78519570041710065450485106721/96428056919040\) | \(1945839794122640409840000000\) | \([2]\) | \(4246732800\) | \(5.4104\) | |
| 381150.fm3 | 381150fm4 | \([1, -1, 0, -158629457817, 20602923190327341]\) | \(21876183941534093095979041/3572502915711058560000\) | \(72090204449999046708176010000000000\) | \([2, 2]\) | \(4246732800\) | \(5.4104\) | |
| 381150.fm6 | 381150fm5 | \([1, -1, 0, 287803914183, 115624927120411341]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-7301171875593052232699285742187500000\) | \([2]\) | \(8493465600\) | \(5.7570\) | |
| 381150.fm2 | 381150fm6 | \([1, -1, 0, -714346157817, -212699728953772659]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(3785316371978007080009637236587500000\) | \([2]\) | \(8493465600\) | \(5.7570\) |
Rank
sage: E.rank()
The elliptic curves in class 381150fm have rank \(0\).
Complex multiplication
The elliptic curves in class 381150fm do not have complex multiplication.Modular form 381150.2.a.fm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
