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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 11550.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11550.a1 | 11550c7 | \([1, 1, 0, -133100025, -591093346125]\) | \(16689299266861680229173649/2396798250\) | \(37449972656250\) | \([2]\) | \(995328\) | \(2.9271\) | |
| 11550.a2 | 11550c8 | \([1, 1, 0, -8537525, -8727158625]\) | \(4404531606962679693649/444872222400201750\) | \(6951128475003152343750\) | \([2]\) | \(995328\) | \(2.9271\) | |
| 11550.a3 | 11550c6 | \([1, 1, 0, -8318775, -9238377375]\) | \(4074571110566294433649/48828650062500\) | \(762947657226562500\) | \([2, 2]\) | \(497664\) | \(2.5806\) | |
| 11550.a4 | 11550c5 | \([1, 1, 0, -1875275, 985693125]\) | \(46676570542430835889/106752955783320\) | \(1668014934114375000\) | \([2]\) | \(331776\) | \(2.3778\) | |
| 11550.a5 | 11550c4 | \([1, 1, 0, -1645275, -809296875]\) | \(31522423139920199089/164434491947880\) | \(2569288936685625000\) | \([2]\) | \(331776\) | \(2.3778\) | |
| 11550.a6 | 11550c3 | \([1, 1, 0, -506275, -152439875]\) | \(-918468938249433649/109183593750000\) | \(-1705993652343750000\) | \([2]\) | \(248832\) | \(2.2340\) | |
| 11550.a7 | 11550c2 | \([1, 1, 0, -160275, 2998125]\) | \(29141055407581489/16604321025600\) | \(259442516025000000\) | \([2, 2]\) | \(165888\) | \(2.0313\) | |
| 11550.a8 | 11550c1 | \([1, 1, 0, 39725, 398125]\) | \(443688652450511/260789760000\) | \(-4074840000000000\) | \([2]\) | \(82944\) | \(1.6847\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 11550.a have rank \(1\).
Complex multiplication
The elliptic curves in class 11550.a do not have complex multiplication.Modular form 11550.2.a.a
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.
