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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 333270eg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.eg7 | 333270eg1 | \([1, -1, 1, -2368697, 1403230169]\) | \(13619385906841/6048000\) | \(652689050313888000\) | \([2]\) | \(9732096\) | \(2.3769\) | \(\Gamma_0(N)\)-optimal |
| 333270.eg6 | 333270eg2 | \([1, -1, 1, -2749577, 921950201]\) | \(21302308926361/8930250000\) | \(963736175854100250000\) | \([2, 2]\) | \(19464192\) | \(2.7234\) | |
| 333270.eg5 | 333270eg3 | \([1, -1, 1, -7010672, -5428185901]\) | \(353108405631241/86318776320\) | \(9315363779854121041920\) | \([2]\) | \(29196288\) | \(2.9262\) | |
| 333270.eg8 | 333270eg4 | \([1, -1, 1, 9152923, 6763697201]\) | \(785793873833639/637994920500\) | \(-68851239875368630060500\) | \([2]\) | \(38928384\) | \(3.0700\) | |
| 333270.eg4 | 333270eg5 | \([1, -1, 1, -20746157, -35726285311]\) | \(9150443179640281/184570312500\) | \(19918489084286132812500\) | \([2]\) | \(38928384\) | \(3.0700\) | |
| 333270.eg2 | 333270eg6 | \([1, -1, 1, -104515952, -411206159149]\) | \(1169975873419524361/108425318400\) | \(11701061193200549990400\) | \([2, 2]\) | \(58392576\) | \(3.2727\) | |
| 333270.eg3 | 333270eg7 | \([1, -1, 1, -96898352, -473694855469]\) | \(-932348627918877961/358766164249920\) | \(-38717385421467698601203520\) | \([2]\) | \(116785152\) | \(3.6193\) | |
| 333270.eg1 | 333270eg8 | \([1, -1, 1, -1672218032, -26319677814061]\) | \(4791901410190533590281/41160000\) | \(4441911592413960000\) | \([2]\) | \(116785152\) | \(3.6193\) |
Rank
sage: E.rank()
The elliptic curves in class 333270eg have rank \(1\).
Complex multiplication
The elliptic curves in class 333270eg do not have complex multiplication.Modular form 333270.2.a.eg
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.
