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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 117810dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117810.df7 | 117810dq1 | \([1, -1, 1, -976279028, -29140823257513]\) | \(-141162084764748587904214427641/421539677967044903067648000\) | \(-307302425237975734336315392000\) | \([2]\) | \(138018816\) | \(4.3450\) | \(\Gamma_0(N)\)-optimal |
| 117810.df6 | 117810dq2 | \([1, -1, 1, -21384661748, -1202312622240169]\) | \(1483553933406627878314880715143161/1904972409734563785924000000\) | \(1388724886696496999938596000000\) | \([2, 2]\) | \(276037632\) | \(4.6915\) | |
| 117810.df8 | 117810dq3 | \([1, -1, 1, 8540736997, 671085211300427]\) | \(94510971880619057444979349412759/321572798571266028122690027520\) | \(-234426570158452934501441030062080\) | \([6]\) | \(414056448\) | \(4.8943\) | |
| 117810.df5 | 117810dq4 | \([1, -1, 1, -27254371748, -489720436704169]\) | \(3071176032738522446354893004903161/1635177816170458876705577958000\) | \(1192044627988264521118366331382000\) | \([2]\) | \(552075264\) | \(5.0381\) | |
| 117810.df3 | 117810dq5 | \([1, -1, 1, -342049075268, -76998144843258793]\) | \(6071016954682394123338855607356153081/10029115297984535156250000\) | \(7311225052230726128906250000\) | \([2]\) | \(552075264\) | \(5.0381\) | |
| 117810.df4 | 117810dq6 | \([1, -1, 1, -82811204123, 7954502395245131]\) | \(86151626782508161683074667552941161/12360692761105045152384575078400\) | \(9010945022845577916088355232153600\) | \([2, 6]\) | \(828112896\) | \(5.2408\) | |
| 117810.df1 | 117810dq7 | \([1, -1, 1, -1275489706523, 554439315123524171]\) | \(314794443646748303921433115102799635561/8206405838866889178408192798720\) | \(5982469856533962211059572550266880\) | \([6]\) | \(1656225792\) | \(5.5874\) | |
| 117810.df2 | 117810dq8 | \([1, -1, 1, -351763759643, -72392706783866293]\) | \(6603124212008881280120689341135103081/715642524575996594697670556160000\) | \(521703400415901517534601835440640000\) | \([6]\) | \(1656225792\) | \(5.5874\) |
Rank
sage: E.rank()
The elliptic curves in class 117810dq have rank \(1\).
Complex multiplication
The elliptic curves in class 117810dq do not have complex multiplication.Modular form 117810.2.a.dq
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.
