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SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 100800.ir
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.ir1 | 100800fw4 | \([0, 0, 0, -2606700, -1619714000]\) | \(2624033547076/324135\) | \(241965480960000000\) | \([2]\) | \(2359296\) | \(2.3589\) | |
| 100800.ir2 | 100800fw2 | \([0, 0, 0, -176700, -20774000]\) | \(3269383504/893025\) | \(166659897600000000\) | \([2, 2]\) | \(1179648\) | \(2.0123\) | |
| 100800.ir3 | 100800fw1 | \([0, 0, 0, -64200, 6001000]\) | \(2508888064/118125\) | \(1377810000000000\) | \([2]\) | \(589824\) | \(1.6657\) | \(\Gamma_0(N)\)-optimal |
| 100800.ir4 | 100800fw3 | \([0, 0, 0, 453300, -135434000]\) | \(13799183324/18600435\) | \(-13885150325760000000\) | \([2]\) | \(2359296\) | \(2.3589\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.ir have rank \(1\).
Complex multiplication
The elliptic curves in class 100800.ir do not have complex multiplication.Modular form 100800.2.a.ir
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
