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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 124950co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 124950.dl7 | 124950co1 | \([1, 0, 1, -39599376, -114328365602]\) | \(-3735772816268612449/909650165760000\) | \(-1672178630492160000000000\) | \([2]\) | \(21233664\) | \(3.3664\) | \(\Gamma_0(N)\)-optimal |
| 124950.dl6 | 124950co2 | \([1, 0, 1, -666799376, -6627173165602]\) | \(17836145204788591940449/770635366502400\) | \(1416632503650638400000000\) | \([2, 2]\) | \(42467328\) | \(3.7130\) | |
| 124950.dl8 | 124950co3 | \([1, 0, 1, 284976624, 777775826398]\) | \(1392333139184610040991/947901937500000000\) | \(-1742495547577148437500000000\) | \([2]\) | \(63700992\) | \(3.9157\) | |
| 124950.dl5 | 124950co4 | \([1, 0, 1, -700119376, -5928252845602]\) | \(20645800966247918737249/3688936444974392640\) | \(6781245059606129995365000000\) | \([2]\) | \(84934656\) | \(4.0596\) | |
| 124950.dl3 | 124950co5 | \([1, 0, 1, -10668679376, -424145651885602]\) | \(73054578035931991395831649/136386452160\) | \(250714526721435000000\) | \([2]\) | \(84934656\) | \(4.0596\) | |
| 124950.dl4 | 124950co6 | \([1, 0, 1, -1246273376, 6486275826398]\) | \(116454264690812369959009/57505157319440250000\) | \(105709753960544155816406250000\) | \([2, 2]\) | \(127401984\) | \(4.2623\) | |
| 124950.dl1 | 124950co7 | \([1, 0, 1, -16292335876, 799835059326398]\) | \(260174968233082037895439009/223081361502731896500\) | \(410082798428670388927007812500\) | \([2]\) | \(254803968\) | \(4.6089\) | |
| 124950.dl2 | 124950co8 | \([1, 0, 1, -10700210876, -421512382673602]\) | \(73704237235978088924479009/899277423164136103500\) | \(1653110774341210131885492187500\) | \([2]\) | \(254803968\) | \(4.6089\) |
Rank
sage: E.rank()
The elliptic curves in class 124950co have rank \(1\).
Complex multiplication
The elliptic curves in class 124950co do not have complex multiplication.Modular form 124950.2.a.co
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.
