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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 34650.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.df1 | 34650da7 | \([1, -1, 1, -1197900230, 15958322445147]\) | \(16689299266861680229173649/2396798250\) | \(27301030066406250\) | \([2]\) | \(7962624\) | \(3.4764\) | |
| 34650.df2 | 34650da8 | \([1, -1, 1, -76837730, 235556445147]\) | \(4404531606962679693649/444872222400201750\) | \(5067372658277298058593750\) | \([2]\) | \(7962624\) | \(3.4764\) | |
| 34650.df3 | 34650da6 | \([1, -1, 1, -74868980, 249361320147]\) | \(4074571110566294433649/48828650062500\) | \(556188842118164062500\) | \([2, 2]\) | \(3981312\) | \(3.1299\) | |
| 34650.df4 | 34650da5 | \([1, -1, 1, -16877480, -26630591853]\) | \(46676570542430835889/106752955783320\) | \(1215982886969379375000\) | \([2]\) | \(2654208\) | \(2.9271\) | |
| 34650.df5 | 34650da4 | \([1, -1, 1, -14807480, 21836208147]\) | \(31522423139920199089/164434491947880\) | \(1873011634843820625000\) | \([2]\) | \(2654208\) | \(2.9271\) | |
| 34650.df6 | 34650da3 | \([1, -1, 1, -4556480, 4111320147]\) | \(-918468938249433649/109183593750000\) | \(-1243669372558593750000\) | \([2]\) | \(1990656\) | \(2.7833\) | |
| 34650.df7 | 34650da2 | \([1, -1, 1, -1442480, -82391853]\) | \(29141055407581489/16604321025600\) | \(189133594182225000000\) | \([2, 2]\) | \(1327104\) | \(2.5806\) | |
| 34650.df8 | 34650da1 | \([1, -1, 1, 357520, -10391853]\) | \(443688652450511/260789760000\) | \(-2970558360000000000\) | \([2]\) | \(663552\) | \(2.2340\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34650.df have rank \(0\).
Complex multiplication
The elliptic curves in class 34650.df do not have complex multiplication.Modular form 34650.2.a.df
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
