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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 397800.cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 397800.cc1 | 397800cc5 | \([0, 0, 0, -43758075, 110552597750]\) | \(397210600760070242/3536192675535\) | \(82492302734880480000000\) | \([2]\) | \(34603008\) | \(3.2204\) | |
| 397800.cc2 | 397800cc3 | \([0, 0, 0, -4743075, -1147347250]\) | \(1011710313226084/536724738225\) | \(6260357346656400000000\) | \([2, 2]\) | \(17301504\) | \(2.8738\) | |
| 397800.cc3 | 397800cc2 | \([0, 0, 0, -3730575, -2770384750]\) | \(1969080716416336/2472575625\) | \(7210030522500000000\) | \([2, 2]\) | \(8650752\) | \(2.5273\) | |
| 397800.cc4 | 397800cc1 | \([0, 0, 0, -3729450, -2772140875]\) | \(31476797652269056/49725\) | \(9062381250000\) | \([2]\) | \(4325376\) | \(2.1807\) | \(\Gamma_0(N)\)-optimal |
| 397800.cc5 | 397800cc4 | \([0, 0, 0, -2736075, -4281030250]\) | \(-194204905090564/566398828125\) | \(-6606475931250000000000\) | \([2]\) | \(17301504\) | \(2.8738\) | |
| 397800.cc6 | 397800cc6 | \([0, 0, 0, 18071925, -8972892250]\) | \(27980756504588158/17683545112935\) | \(-412521740394547680000000\) | \([2]\) | \(34603008\) | \(3.2204\) |
Rank
sage: E.rank()
The elliptic curves in class 397800.cc have rank \(0\).
Complex multiplication
The elliptic curves in class 397800.cc do not have complex multiplication.Modular form 397800.2.a.cc
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
