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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 212160.cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.cc1 | 212160gf8 | \([0, -1, 0, -421824000385, -105449485114502783]\) | \(31664865542564944883878115208137569/103216295812500\) | \(27057532649472000000\) | \([2]\) | \(637009920\) | \(4.8139\) | |
| 212160.cc2 | 212160gf6 | \([0, -1, 0, -26364000385, -1647641566502783]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(115702550406144000000000000\) | \([2, 2]\) | \(318504960\) | \(4.4673\) | |
| 212160.cc3 | 212160gf7 | \([0, -1, 0, -26316193665, -1653914697371775]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-15315424312500000000000000000000\) | \([2]\) | \(637009920\) | \(4.8139\) | |
| 212160.cc4 | 212160gf5 | \([0, -1, 0, -5207889025, -144637146850175]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(2315889543102414715669708800\) | \([2]\) | \(212336640\) | \(4.2646\) | |
| 212160.cc5 | 212160gf3 | \([0, -1, 0, -1650738305, -25645921710975]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(3737761374524767469568000000\) | \([2]\) | \(159252480\) | \(4.1208\) | |
| 212160.cc6 | 212160gf2 | \([0, -1, 0, -355665025, -1815815189375]\) | \(18980483520595353274840609/5549773448629762560000\) | \(1454839810917600476528640000\) | \([2, 2]\) | \(106168320\) | \(3.9180\) | |
| 212160.cc7 | 212160gf1 | \([0, -1, 0, -134153345, 575935024257]\) | \(1018563973439611524445729/42904970360310988800\) | \(11247280550133363847987200\) | \([2]\) | \(53084160\) | \(3.5715\) | \(\Gamma_0(N)\)-optimal |
| 212160.cc8 | 212160gf4 | \([0, -1, 0, 952372095, -12070041387903]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-118217061600493495910400000000\) | \([2]\) | \(212336640\) | \(4.2646\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.cc have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.cc do not have complex multiplication.Modular form 212160.2.a.cc
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.
