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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 55440.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.q1 | 55440cq8 | \([0, 0, 0, -36925563, 81031303082]\) | \(1864737106103260904761/129177711985836360\) | \(385722581146315597578240\) | \([2]\) | \(5308416\) | \(3.2737\) | |
| 55440.q2 | 55440cq5 | \([0, 0, 0, -36288363, 84139355162]\) | \(1769857772964702379561/691787250\) | \(2065665659904000\) | \([2]\) | \(1769472\) | \(2.7244\) | |
| 55440.q3 | 55440cq6 | \([0, 0, 0, -7290363, -6054695638]\) | \(14351050585434661561/3001282273281600\) | \(8961780847502485094400\) | \([2, 2]\) | \(2654208\) | \(2.9272\) | |
| 55440.q4 | 55440cq3 | \([0, 0, 0, -6875643, -6938961622]\) | \(12038605770121350841/757333463040\) | \(2261385603302031360\) | \([2]\) | \(1327104\) | \(2.5806\) | |
| 55440.q5 | 55440cq2 | \([0, 0, 0, -2268363, 1314263162]\) | \(432288716775559561/270140062500\) | \(806633904384000000\) | \([2, 2]\) | \(884736\) | \(2.3779\) | |
| 55440.q6 | 55440cq4 | \([0, 0, 0, -1840683, 1824998618]\) | \(-230979395175477481/348191894531250\) | \(-1039695426000000000000\) | \([2]\) | \(1769472\) | \(2.7244\) | |
| 55440.q7 | 55440cq1 | \([0, 0, 0, -168843, 12140858]\) | \(178272935636041/81841914000\) | \(244378645733376000\) | \([2]\) | \(442368\) | \(2.0313\) | \(\Gamma_0(N)\)-optimal |
| 55440.q8 | 55440cq7 | \([0, 0, 0, 15709317, -36547671382]\) | \(143584693754978072519/276341298967965000\) | \(-825150697257560002560000\) | \([2]\) | \(5308416\) | \(3.2737\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.q have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.q do not have complex multiplication.Modular form 55440.2.a.q
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
