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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 193200cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.dy4 | 193200cb1 | \([0, 1, 0, -304008, -64608012]\) | \(48551226272641/9273600\) | \(593510400000000\) | \([2]\) | \(1179648\) | \(1.8356\) | \(\Gamma_0(N)\)-optimal |
| 193200.dy3 | 193200cb2 | \([0, 1, 0, -336008, -50208012]\) | \(65553197996161/20996010000\) | \(1343744640000000000\) | \([2, 2]\) | \(2359296\) | \(2.1821\) | |
| 193200.dy2 | 193200cb3 | \([0, 1, 0, -2136008, 1162991988]\) | \(16840406336564161/604708416900\) | \(38701338681600000000\) | \([2, 2]\) | \(4718592\) | \(2.5287\) | |
| 193200.dy6 | 193200cb4 | \([0, 1, 0, 951992, -341296012]\) | \(1490881681033919/1650501562500\) | \(-105632100000000000000\) | \([2]\) | \(4718592\) | \(2.5287\) | |
| 193200.dy1 | 193200cb5 | \([0, 1, 0, -33876008, 75878951988]\) | \(67176973097223766561/91487391870\) | \(5855193079680000000\) | \([2]\) | \(9437184\) | \(2.8753\) | |
| 193200.dy5 | 193200cb6 | \([0, 1, 0, 803992, 4120631988]\) | \(898045580910239/115117148363070\) | \(-7367497495236480000000\) | \([2]\) | \(9437184\) | \(2.8753\) |
Rank
sage: E.rank()
The elliptic curves in class 193200cb have rank \(1\).
Complex multiplication
The elliptic curves in class 193200cb do not have complex multiplication.Modular form 193200.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
