Show commands:
SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 303450.cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.cw1 | 303450cw8 | \([1, 0, 1, -96091532001, -11456557036270352]\) | \(260174968233082037895439009/223081361502731896500\) | \(84135027435720856879212632812500\) | \([2]\) | \(1528823808\) | \(5.0525\) | |
| 303450.cw2 | 303450cw7 | \([1, 0, 1, -63109407001, 6037563123479648]\) | \(73704237235978088924479009/899277423164136103500\) | \(339162044558852086306599867187500\) | \([2]\) | \(1528823808\) | \(5.0525\) | |
| 303450.cw3 | 303450cw4 | \([1, 0, 1, -62923435501, 6075281111420648]\) | \(73054578035931991395831649/136386452160\) | \(51438084369956235000000\) | \([2]\) | \(509607936\) | \(4.5032\) | |
| 303450.cw4 | 303450cw6 | \([1, 0, 1, -7350469501, -92909018895352]\) | \(116454264690812369959009/57505157319440250000\) | \(21688042228966313683628906250000\) | \([2, 2]\) | \(764411904\) | \(4.7060\) | |
| 303450.cw5 | 303450cw5 | \([1, 0, 1, -4129275501, 84912832540648]\) | \(20645800966247918737249/3688936444974392640\) | \(1391280593393501649704565000000\) | \([2]\) | \(509607936\) | \(4.5032\) | |
| 303450.cw6 | 303450cw2 | \([1, 0, 1, -3932755501, 94923954380648]\) | \(17836145204788591940449/770635366502400\) | \(290644755199874510400000000\) | \([2, 2]\) | \(254803968\) | \(4.1567\) | |
| 303450.cw7 | 303450cw1 | \([1, 0, 1, -233555501, 1637528780648]\) | \(-3735772816268612449/909650165760000\) | \(-343074119404584960000000000\) | \([2]\) | \(127401984\) | \(3.8101\) | \(\Gamma_0(N)\)-optimal |
| 303450.cw8 | 303450cw3 | \([1, 0, 1, 1680780499, -11140081395352]\) | \(1392333139184610040991/947901937500000000\) | \(-357500756588124023437500000000\) | \([2]\) | \(382205952\) | \(4.3594\) |
Rank
sage: E.rank()
The elliptic curves in class 303450.cw have rank \(1\).
Complex multiplication
The elliptic curves in class 303450.cw do not have complex multiplication.Modular form 303450.2.a.cw
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.
