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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 177870.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.cp1 | 177870ge6 | \([1, 0, 1, -99607324, -382643588428]\) | \(524388516989299201/3150\) | \(656530497280350\) | \([2]\) | \(15728640\) | \(2.9087\) | |
| 177870.cp2 | 177870ge4 | \([1, 0, 1, -6225574, -5978961628]\) | \(128031684631201/9922500\) | \(2068071066433102500\) | \([2, 2]\) | \(7864320\) | \(2.5621\) | |
| 177870.cp3 | 177870ge5 | \([1, 0, 1, -5810544, -6810349724]\) | \(-104094944089921/35880468750\) | \(-7478292695583986718750\) | \([2]\) | \(15728640\) | \(2.9087\) | |
| 177870.cp4 | 177870ge3 | \([1, 0, 1, -2193854, 1181942276]\) | \(5602762882081/345888060\) | \(72090812709566837340\) | \([2]\) | \(7864320\) | \(2.5621\) | |
| 177870.cp5 | 177870ge2 | \([1, 0, 1, -415154, -80223244]\) | \(37966934881/8643600\) | \(1801519684537280400\) | \([2, 2]\) | \(3932160\) | \(2.2155\) | |
| 177870.cp6 | 177870ge1 | \([1, 0, 1, 59166, -7747148]\) | \(109902239/188160\) | \(-39216755037546240\) | \([2]\) | \(1966080\) | \(1.8689\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 177870.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 177870.cp do not have complex multiplication.Modular form 177870.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
