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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 31878v
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31878.u6 | 31878v1 | \([1, -1, 0, 937944, -45130176]\) | \(125177609053596564863/73635189229502208\) | \(-53680052948307109632\) | \([2]\) | \(819200\) | \(2.4752\) | \(\Gamma_0(N)\)-optimal |
| 31878.u5 | 31878v2 | \([1, -1, 0, -3785976, -359743248]\) | \(8232463578739844255617/4687062591766850064\) | \(3416868629398033696656\) | \([2, 2]\) | \(1638400\) | \(2.8218\) | |
| 31878.u3 | 31878v3 | \([1, -1, 0, -38792556, 92582726652]\) | \(8856076866003496152467137/46664863048067576004\) | \(34018685162041262906916\) | \([2, 2]\) | \(3276800\) | \(3.1683\) | |
| 31878.u2 | 31878v4 | \([1, -1, 0, -44362116, -113494136796]\) | \(13244420128496241770842177/29965867631164664892\) | \(21845117503119040706268\) | \([2]\) | \(3276800\) | \(3.1683\) | |
| 31878.u4 | 31878v5 | \([1, -1, 0, -17796546, 192393558990]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-15629128000948327594857282\) | \([2]\) | \(6553600\) | \(3.5149\) | |
| 31878.u1 | 31878v6 | \([1, -1, 0, -619893846, 5940669888954]\) | \(36136672427711016379227705697/1011258101510224722\) | \(737207156000953822338\) | \([2]\) | \(6553600\) | \(3.5149\) |
Rank
sage: E.rank()
The elliptic curves in class 31878v have rank \(1\).
Complex multiplication
The elliptic curves in class 31878v do not have complex multiplication.Modular form 31878.2.a.v
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.
