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SageMath
E = EllipticCurve("nk1")
E.isogeny_class()
Elliptic curves in class 242550.nk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
242550.nk1 | 242550nk8 | \([1, -1, 1, -58697111255, -5473587204463003]\) | \(16689299266861680229173649/2396798250\) | \(3211938886282628906250\) | \([2]\) | \(382205952\) | \(4.4494\) | |
242550.nk2 | 242550nk7 | \([1, -1, 1, -3765048755, -80788330588003]\) | \(4404531606962679693649/444872222400201750\) | \(596171325873665839295496093750\) | \([2]\) | \(382205952\) | \(4.4494\) | |
242550.nk3 | 242550nk6 | \([1, -1, 1, -3668580005, -85523595650503]\) | \(4074571110566294433649/48828650062500\) | \(65435061086359883789062500\) | \([2, 2]\) | \(191102976\) | \(4.1028\) | |
242550.nk4 | 242550nk4 | \([1, -1, 1, -826996505, 9135946998497]\) | \(46676570542430835889/106752955783320\) | \(143059170669060514089375000\) | \([2]\) | \(127401984\) | \(3.9001\) | |
242550.nk5 | 242550nk5 | \([1, -1, 1, -725566505, -7488368261503]\) | \(31522423139920199089/164434491947880\) | \(220357945827740652710625000\) | \([2]\) | \(127401984\) | \(3.9001\) | |
242550.nk6 | 242550nk3 | \([1, -1, 1, -223267505, -1409736275503]\) | \(-918468938249433649/109183593750000\) | \(-146316458012145996093750000\) | \([2]\) | \(95551488\) | \(3.7563\) | |
242550.nk7 | 242550nk2 | \([1, -1, 1, -70681505, 28401768497]\) | \(29141055407581489/16604321025600\) | \(22251378221944589025000000\) | \([2, 2]\) | \(63700992\) | \(3.5535\) | |
242550.nk8 | 242550nk1 | \([1, -1, 1, 17518495, 3529368497]\) | \(443688652450511/260789760000\) | \(-349483220495640000000000\) | \([2]\) | \(31850496\) | \(3.2069\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 242550.nk have rank \(1\).
Complex multiplication
The elliptic curves in class 242550.nk do not have complex multiplication.Modular form 242550.2.a.nk
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.