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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 54978.cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54978.cc1 | 54978bw6 | \([1, 0, 0, -19352257, 32766056357]\) | \(6812873765474836663297/74052\) | \(8712143748\) | \([2]\) | \(1572864\) | \(2.4109\) | |
| 54978.cc2 | 54978bw4 | \([1, 0, 0, -1209517, 511893185]\) | \(1663303207415737537/5483698704\) | \(645151668826896\) | \([2, 2]\) | \(786432\) | \(2.0643\) | |
| 54978.cc3 | 54978bw5 | \([1, 0, 0, -1192857, 526683933]\) | \(-1595514095015181697/95635786040388\) | \(-11251454591865607812\) | \([2]\) | \(1572864\) | \(2.4109\) | |
| 54978.cc4 | 54978bw2 | \([1, 0, 0, -76637, 7761585]\) | \(423108074414017/23284318464\) | \(2739376782971136\) | \([2, 2]\) | \(393216\) | \(1.7177\) | |
| 54978.cc5 | 54978bw1 | \([1, 0, 0, -13917, -479823]\) | \(2533811507137/625016832\) | \(73532605267968\) | \([2]\) | \(196608\) | \(1.3712\) | \(\Gamma_0(N)\)-optimal |
| 54978.cc6 | 54978bw3 | \([1, 0, 0, 52723, 31330977]\) | \(137763859017023/3683199928848\) | \(-433324788429038352\) | \([2]\) | \(786432\) | \(2.0643\) |
Rank
sage: E.rank()
The elliptic curves in class 54978.cc have rank \(0\).
Complex multiplication
The elliptic curves in class 54978.cc do not have complex multiplication.Modular form 54978.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
