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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 44100cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44100.b2 | 44100cp1 | \([0, 0, 0, -676200, 213474625]\) | \(1594753024/4725\) | \(101311230431250000\) | \([2]\) | \(663552\) | \(2.1329\) | \(\Gamma_0(N)\)-optimal |
44100.b3 | 44100cp2 | \([0, 0, 0, -400575, 389047750]\) | \(-20720464/178605\) | \(-61273032164820000000\) | \([2]\) | \(1327104\) | \(2.4795\) | |
44100.b1 | 44100cp3 | \([0, 0, 0, -3322200, -2152710875]\) | \(189123395584/16078125\) | \(344739603550781250000\) | \([2]\) | \(1990656\) | \(2.6822\) | |
44100.b4 | 44100cp4 | \([0, 0, 0, 3568425, -9918445250]\) | \(14647977776/132355125\) | \(-45406342662880500000000\) | \([2]\) | \(3981312\) | \(3.0288\) |
Rank
sage: E.rank()
The elliptic curves in class 44100cp have rank \(2\).
Complex multiplication
The elliptic curves in class 44100cp do not have complex multiplication.Modular form 44100.2.a.cp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.