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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 196350.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 196350.m1 | 196350fy8 | \([1, 1, 0, -33333883333900, -74075921305978583000]\) | \(262156976355489363181342849900999019467969/296485141924125000\) | \(4632580342564453125000\) | \([2]\) | \(5350883328\) | \(5.8283\) | |
| 196350.m2 | 196350fy6 | \([1, 1, 0, -2083367708900, -1157436920837958000]\) | \(64003168104546012500462338813649467969/68064746081030015625000000\) | \(1063511657516093994140625000000\) | \([2, 2]\) | \(2675441664\) | \(5.4818\) | |
| 196350.m3 | 196350fy7 | \([1, 1, 0, -2082852083900, -1158038473197333000]\) | \(-63955658296770964115513956628279467969/66004356107812185925891924125000\) | \(-1031318064184565405092061314453125000\) | \([2]\) | \(5350883328\) | \(5.8283\) | |
| 196350.m4 | 196350fy5 | \([1, 1, 0, -411530452900, -101612679621950000]\) | \(493298302018650738343048153196947009/5139490792463830279120089600\) | \(80304543632247348111251400000000\) | \([2]\) | \(1783627776\) | \(5.2790\) | |
| 196350.m5 | 196350fy3 | \([1, 1, 0, -130242708900, -18075592712958000]\) | \(15637378471582822120727563649467969/16113547119140625000000000000\) | \(251774173736572265625000000000000\) | \([2]\) | \(1337720832\) | \(5.1352\) | |
| 196350.m6 | 196350fy2 | \([1, 1, 0, -26351252900, -1505761079550000]\) | \(129511249478743944259581330835009/12262789317997149185802240000\) | \(191606083093705456028160000000000\) | \([2, 2]\) | \(891813888\) | \(4.9324\) | |
| 196350.m7 | 196350fy1 | \([1, 1, 0, -5871252900, 146913480450000]\) | \(1432504679512464302827718035009/232233326153721446400000000\) | \(3628645721151897600000000000000\) | \([2]\) | \(445906944\) | \(4.5859\) | \(\Gamma_0(N)\)-optimal |
| 196350.m8 | 196350fy4 | \([1, 1, 0, 31147947100, -7169604777150000]\) | \(213890734289719241265598586476991/1544981081981970035652027609600\) | \(-24140329405968281807062931400000000\) | \([2]\) | \(1783627776\) | \(5.2790\) |
Rank
sage: E.rank()
The elliptic curves in class 196350.m have rank \(0\).
Complex multiplication
The elliptic curves in class 196350.m do not have complex multiplication.Modular form 196350.2.a.m
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
