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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 382200n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 382200.n6 | 382200n1 | \([0, -1, 0, 882817, 1299321612]\) | \(2587063175168/26304786963\) | \(-773682970352496750000\) | \([2]\) | \(14155776\) | \(2.6892\) | \(\Gamma_0(N)\)-optimal |
| 382200.n5 | 382200n2 | \([0, -1, 0, -13823308, 18358426612]\) | \(620742479063632/49991146569\) | \(23525633610785124000000\) | \([2, 2]\) | \(28311552\) | \(3.0358\) | |
| 382200.n2 | 382200n3 | \([0, -1, 0, -216707808, 1227955815612]\) | \(597914615076708388/4400862921\) | \(8284113948683664000000\) | \([2, 2]\) | \(56623104\) | \(3.3824\) | |
| 382200.n4 | 382200n4 | \([0, -1, 0, -46236808, -99691540388]\) | \(5807363790481348/1079211743883\) | \(2031490919297457072000000\) | \([2]\) | \(56623104\) | \(3.3824\) | |
| 382200.n1 | 382200n5 | \([0, -1, 0, -3467318808, 78585996393612]\) | \(1224522642327678150914/66339\) | \(249750944352000000\) | \([2]\) | \(113246208\) | \(3.7290\) | |
| 382200.n3 | 382200n6 | \([0, -1, 0, -212248808, 1280901981612]\) | \(-280880296871140514/25701087819771\) | \(-96758632989063628128000000\) | \([2]\) | \(113246208\) | \(3.7290\) |
Rank
sage: E.rank()
The elliptic curves in class 382200n have rank \(1\).
Complex multiplication
The elliptic curves in class 382200n do not have complex multiplication.Modular form 382200.2.a.n
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.
