Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 17850.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17850.e1 | 17850b7 | \([1, 1, 0, -332496650, -2332023138000]\) | \(260174968233082037895439009/223081361502731896500\) | \(3485646273480185882812500\) | \([2]\) | \(5308416\) | \(3.6359\) | |
| 17850.e2 | 17850b8 | \([1, 1, 0, -218371650, 1228805487000]\) | \(73704237235978088924479009/899277423164136103500\) | \(14051209736939626617187500\) | \([4]\) | \(5308416\) | \(3.6359\) | |
| 17850.e3 | 17850b5 | \([1, 1, 0, -217728150, 1236482932500]\) | \(73054578035931991395831649/136386452160\) | \(2131038315000000\) | \([4]\) | \(1769472\) | \(3.0866\) | |
| 17850.e4 | 17850b6 | \([1, 1, 0, -25434150, -18921325500]\) | \(116454264690812369959009/57505157319440250000\) | \(898518083116253906250000\) | \([2, 2]\) | \(2654208\) | \(3.2894\) | |
| 17850.e5 | 17850b4 | \([1, 1, 0, -14288150, 17277412500]\) | \(20645800966247918737249/3688936444974392640\) | \(57639631952724885000000\) | \([2]\) | \(1769472\) | \(3.0866\) | |
| 17850.e6 | 17850b2 | \([1, 1, 0, -13608150, 19315372500]\) | \(17836145204788591940449/770635366502400\) | \(12041177601600000000\) | \([2, 2]\) | \(884736\) | \(2.7401\) | |
| 17850.e7 | 17850b1 | \([1, 1, 0, -808150, 332972500]\) | \(-3735772816268612449/909650165760000\) | \(-14213283840000000000\) | \([2]\) | \(442368\) | \(2.3935\) | \(\Gamma_0(N)\)-optimal |
| 17850.e8 | 17850b3 | \([1, 1, 0, 5815850, -2265075500]\) | \(1392333139184610040991/947901937500000000\) | \(-14810967773437500000000\) | \([2]\) | \(1327104\) | \(2.9428\) |
Rank
sage: E.rank()
The elliptic curves in class 17850.e have rank \(0\).
Complex multiplication
The elliptic curves in class 17850.e do not have complex multiplication.Modular form 17850.2.a.e
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
