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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 210210.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 210210.bd1 | 210210ev7 | \([1, 1, 0, -5181876102, 143568983020716]\) | \(130796627670002750950880364889/4007004103295286093000\) | \(471420025748587113555357000\) | \([2]\) | \(286654464\) | \(4.2156\) | |
| 210210.bd2 | 210210ev6 | \([1, 1, 0, -337491102, 2044150753716]\) | \(36134533748915083453404889/5565686539253841000000\) | \(654797455656675139809000000\) | \([2, 2]\) | \(143327232\) | \(3.8690\) | |
| 210210.bd3 | 210210ev4 | \([1, 1, 0, -113381517, -146515113261]\) | \(1370131553911340548947529/714126686285699857170\) | \(84016290514826302496193330\) | \([2]\) | \(95551488\) | \(3.6663\) | |
| 210210.bd4 | 210210ev3 | \([1, 1, 0, -92491102, -311328246284]\) | \(743764321292317933404889/74603529000000000000\) | \(8777030583321000000000000\) | \([2]\) | \(71663616\) | \(3.5224\) | |
| 210210.bd5 | 210210ev2 | \([1, 1, 0, -90162867, -329241245031]\) | \(688999042618248810121129/779639711718968100\) | \(91723832444024877996900\) | \([2, 2]\) | \(47775744\) | \(3.3197\) | |
| 210210.bd6 | 210210ev1 | \([1, 1, 0, -90138367, -329429253131]\) | \(688437529087783927489129/882972090000\) | \(103880783416410000\) | \([2]\) | \(23887872\) | \(2.9731\) | \(\Gamma_0(N)\)-optimal |
| 210210.bd7 | 210210ev5 | \([1, 1, 0, -67336217, -499934368401]\) | \(-286999819333751016766729/751553009101890965970\) | \(-88419459967828370255404530\) | \([2]\) | \(95551488\) | \(3.6663\) | |
| 210210.bd8 | 210210ev8 | \([1, 1, 0, 586893898, 11274874486716]\) | \(190026536708029086053555111/576736012771479654093000\) | \(-67852415166551809824387357000\) | \([2]\) | \(286654464\) | \(4.2156\) |
Rank
sage: E.rank()
The elliptic curves in class 210210.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 210210.bd do not have complex multiplication.Modular form 210210.2.a.bd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
