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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 67600.cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67600.cb1 | 67600b4 | \([0, 0, 0, -452075, 116990250]\) | \(132304644/5\) | \(386144720000000\) | \([2]\) | \(442368\) | \(1.8850\) | |
67600.cb2 | 67600b2 | \([0, 0, 0, -29575, 1647750]\) | \(148176/25\) | \(482680900000000\) | \([2, 2]\) | \(221184\) | \(1.5384\) | |
67600.cb3 | 67600b1 | \([0, 0, 0, -8450, -274625]\) | \(55296/5\) | \(6033511250000\) | \([2]\) | \(110592\) | \(1.1918\) | \(\Gamma_0(N)\)-optimal |
67600.cb4 | 67600b3 | \([0, 0, 0, 54925, 9337250]\) | \(237276/625\) | \(-48268090000000000\) | \([2]\) | \(442368\) | \(1.8850\) |
Rank
sage: E.rank()
The elliptic curves in class 67600.cb have rank \(1\).
Complex multiplication
The elliptic curves in class 67600.cb do not have complex multiplication.Modular form 67600.2.a.cb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.