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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 18240x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18240.bm4 | 18240x1 | \([0, -1, 0, 235295, -263738015]\) | \(5495662324535111/117739817533440\) | \(-30864786727486095360\) | \([2]\) | \(430080\) | \(2.4200\) | \(\Gamma_0(N)\)-optimal |
| 18240.bm3 | 18240x2 | \([0, -1, 0, -5007585, -4083700383]\) | \(52974743974734147769/3152005008998400\) | \(826279201078876569600\) | \([2, 2]\) | \(860160\) | \(2.7666\) | |
| 18240.bm1 | 18240x3 | \([0, -1, 0, -78940385, -269931262623]\) | \(207530301091125281552569/805586668007040\) | \(211179711498037493760\) | \([2]\) | \(1720320\) | \(3.1131\) | |
| 18240.bm2 | 18240x4 | \([0, -1, 0, -14960865, 17202384225]\) | \(1412712966892699019449/330160465517040000\) | \(86549585072498933760000\) | \([4]\) | \(1720320\) | \(3.1131\) |
Rank
sage: E.rank()
The elliptic curves in class 18240x have rank \(1\).
Complex multiplication
The elliptic curves in class 18240x do not have complex multiplication.Modular form 18240.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
