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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 12240.ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12240.ce1 | 12240cc3 | \([0, 0, 0, -941907, -351847406]\) | \(30949975477232209/478125000\) | \(1427673600000000\) | \([2]\) | \(147456\) | \(2.0430\) | |
12240.ce2 | 12240cc2 | \([0, 0, 0, -60627, -5151854]\) | \(8253429989329/936360000\) | \(2795955978240000\) | \([2, 2]\) | \(73728\) | \(1.6964\) | |
12240.ce3 | 12240cc1 | \([0, 0, 0, -14547, 589714]\) | \(114013572049/15667200\) | \(46782008524800\) | \([2]\) | \(36864\) | \(1.3498\) | \(\Gamma_0(N)\)-optimal |
12240.ce4 | 12240cc4 | \([0, 0, 0, 83373, -25916654]\) | \(21464092074671/109596256200\) | \(-327252667473100800\) | \([2]\) | \(147456\) | \(2.0430\) |
Rank
sage: E.rank()
The elliptic curves in class 12240.ce have rank \(0\).
Complex multiplication
The elliptic curves in class 12240.ce do not have complex multiplication.Modular form 12240.2.a.ce
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.