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SageMath
E = EllipticCurve("on1")
E.isogeny_class()
Elliptic curves in class 374850.on
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
374850.on1 | 374850on4 | \([1, -1, 1, -72114755, -235692245253]\) | \(30949975477232209/478125000\) | \(640733228173828125000\) | \([2]\) | \(42467328\) | \(3.1275\) | |
374850.on2 | 374850on2 | \([1, -1, 1, -4641755, -3450179253]\) | \(8253429989329/936360000\) | \(1254811954055625000000\) | \([2, 2]\) | \(21233664\) | \(2.7809\) | |
374850.on3 | 374850on1 | \([1, -1, 1, -1113755, 395340747]\) | \(114013572049/15667200\) | \(20995546420800000000\) | \([2]\) | \(10616832\) | \(2.4344\) | \(\Gamma_0(N)\)-optimal |
374850.on4 | 374850on3 | \([1, -1, 1, 6383245, -17363729253]\) | \(21464092074671/109596256200\) | \(-146869465162440628125000\) | \([2]\) | \(42467328\) | \(3.1275\) |
Rank
sage: E.rank()
The elliptic curves in class 374850.on have rank \(0\).
Complex multiplication
The elliptic curves in class 374850.on do not have complex multiplication.Modular form 374850.2.a.on
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.