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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 10890bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10890.bi6 | 10890bo1 | \([1, -1, 1, 277672, 5282907]\) | \(1833318007919/1070530560\) | \(-1382555928075632640\) | \([2]\) | \(184320\) | \(2.1703\) | \(\Gamma_0(N)\)-optimal |
| 10890.bi5 | 10890bo2 | \([1, -1, 1, -1116248, 43197531]\) | \(119102750067601/68309049600\) | \(88218949551232262400\) | \([2, 2]\) | \(368640\) | \(2.5169\) | |
| 10890.bi3 | 10890bo3 | \([1, -1, 1, -11657768, -15254656293]\) | \(135670761487282321/643043610000\) | \(830470224985128090000\) | \([2, 2]\) | \(737280\) | \(2.8634\) | |
| 10890.bi2 | 10890bo4 | \([1, -1, 1, -12877448, 17750860251]\) | \(182864522286982801/463015182960\) | \(597969277953514608240\) | \([2]\) | \(737280\) | \(2.8634\) | |
| 10890.bi1 | 10890bo5 | \([1, -1, 1, -186311588, -978784850469]\) | \(553808571467029327441/12529687500\) | \(16181690067829687500\) | \([2]\) | \(1474560\) | \(3.2100\) | |
| 10890.bi4 | 10890bo6 | \([1, -1, 1, -5668268, -30913605093]\) | \(-15595206456730321/310672490129100\) | \(-401223569851201324797900\) | \([2]\) | \(1474560\) | \(3.2100\) |
Rank
sage: E.rank()
The elliptic curves in class 10890bo have rank \(0\).
Complex multiplication
The elliptic curves in class 10890bo do not have complex multiplication.Modular form 10890.2.a.bo
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.
