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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 179088cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 179088.h4 | 179088cc1 | \([0, -1, 0, -130319, -18061890]\) | \(15297845776153065472/2239953782157\) | \(35839260514512\) | \([2]\) | \(774144\) | \(1.6153\) | \(\Gamma_0(N)\)-optimal |
| 179088.h3 | 179088cc2 | \([0, -1, 0, -142324, -14522816]\) | \(1245435444557269072/362494677939849\) | \(92798637552601344\) | \([2, 2]\) | \(1548288\) | \(1.9619\) | |
| 179088.h2 | 179088cc3 | \([0, -1, 0, -856744, 294106624]\) | \(67916840974723357348/2882919026938041\) | \(2952109083584553984\) | \([2, 4]\) | \(3096576\) | \(2.3084\) | |
| 179088.h5 | 179088cc4 | \([0, -1, 0, 380016, -96843600]\) | \(5926909004619535292/7376212204274277\) | \(-7553241297176859648\) | \([2]\) | \(3096576\) | \(2.3084\) | |
| 179088.h1 | 179088cc5 | \([0, -1, 0, -13565104, 19234646368]\) | \(134791707552592102058594/259164991638189\) | \(530769902875011072\) | \([4]\) | \(6193152\) | \(2.6550\) | |
| 179088.h6 | 179088cc6 | \([0, -1, 0, 420896, 1094420320]\) | \(4026401908965753406/255046496743353861\) | \(-522335225330388707328\) | \([4]\) | \(6193152\) | \(2.6550\) |
Rank
sage: E.rank()
The elliptic curves in class 179088cc have rank \(1\).
Complex multiplication
The elliptic curves in class 179088cc do not have complex multiplication.Modular form 179088.2.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 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.
