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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 142800bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 142800.hz7 | 142800bi1 | \([0, 1, 0, -12930408, -21336100812]\) | \(-3735772816268612449/909650165760000\) | \(-58217610608640000000000\) | \([2]\) | \(10616832\) | \(3.0866\) | \(\Gamma_0(N)\)-optimal |
| 142800.hz6 | 142800bi2 | \([0, 1, 0, -217730408, -1236619300812]\) | \(17836145204788591940449/770635366502400\) | \(49320663456153600000000\) | \([2, 2]\) | \(21233664\) | \(3.4332\) | |
| 142800.hz8 | 142800bi3 | \([0, 1, 0, 93053592, 145150939188]\) | \(1392333139184610040991/947901937500000000\) | \(-60665724000000000000000000\) | \([2]\) | \(31850496\) | \(3.6359\) | |
| 142800.hz5 | 142800bi4 | \([0, 1, 0, -228610408, -1106211620812]\) | \(20645800966247918737249/3688936444974392640\) | \(236091932478361128960000000\) | \([4]\) | \(42467328\) | \(3.7798\) | |
| 142800.hz3 | 142800bi5 | \([0, 1, 0, -3483650408, -79141874980812]\) | \(73054578035931991395831649/136386452160\) | \(8728732938240000000\) | \([2]\) | \(42467328\) | \(3.7798\) | |
| 142800.hz4 | 142800bi6 | \([0, 1, 0, -406946408, 1210150939188]\) | \(116454264690812369959009/57505157319440250000\) | \(3680330068444176000000000000\) | \([2, 2]\) | \(63700992\) | \(3.9825\) | |
| 142800.hz1 | 142800bi7 | \([0, 1, 0, -5319946408, 149238840939188]\) | \(260174968233082037895439009/223081361502731896500\) | \(14277207136174841376000000000\) | \([4]\) | \(127401984\) | \(4.3291\) | |
| 142800.hz2 | 142800bi8 | \([0, 1, 0, -3493946408, -78650539060812]\) | \(73704237235978088924479009/899277423164136103500\) | \(57553755082504710624000000000\) | \([2]\) | \(127401984\) | \(4.3291\) |
Rank
sage: E.rank()
The elliptic curves in class 142800bi have rank \(1\).
Complex multiplication
The elliptic curves in class 142800bi do not have complex multiplication.Modular form 142800.2.a.bi
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.
