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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 55440.cx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.cx1 | 55440dz7 | \([0, 0, 0, -766656147, -8170507760686]\) | \(16689299266861680229173649/2396798250\) | \(7156801225728000\) | \([2]\) | \(7962624\) | \(3.3649\) | |
55440.cx2 | 55440dz8 | \([0, 0, 0, -49176147, -120595064686]\) | \(4404531606962679693649/444872222400201750\) | \(1328381338131444022272000\) | \([2]\) | \(7962624\) | \(3.3649\) | |
55440.cx3 | 55440dz6 | \([0, 0, 0, -47916147, -127663412686]\) | \(4074571110566294433649/48828650062500\) | \(145801567828224000000\) | \([2, 2]\) | \(3981312\) | \(3.0183\) | |
55440.cx4 | 55440dz5 | \([0, 0, 0, -10801587, 13637023346]\) | \(46676570542430835889/106752955783320\) | \(318762617921700986880\) | \([2]\) | \(2654208\) | \(2.8156\) | |
55440.cx5 | 55440dz4 | \([0, 0, 0, -9476787, -11178243214]\) | \(31522423139920199089/164434491947880\) | \(490998762004498513920\) | \([2]\) | \(2654208\) | \(2.8156\) | |
55440.cx6 | 55440dz3 | \([0, 0, 0, -2916147, -2104412686]\) | \(-918468938249433649/109183593750000\) | \(-326020464000000000000\) | \([2]\) | \(1990656\) | \(2.6717\) | |
55440.cx7 | 55440dz2 | \([0, 0, 0, -923187, 42369266]\) | \(29141055407581489/16604321025600\) | \(49580236913305190400\) | \([2, 2]\) | \(1327104\) | \(2.4690\) | |
55440.cx8 | 55440dz1 | \([0, 0, 0, 228813, 5274866]\) | \(443688652450511/260789760000\) | \(-778714050723840000\) | \([2]\) | \(663552\) | \(2.1224\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55440.cx have rank \(0\).
Complex multiplication
The elliptic curves in class 55440.cx do not have complex multiplication.Modular form 55440.2.a.cx
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.