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SageMath
E = EllipticCurve("gq1")
E.isogeny_class()
Elliptic curves in class 199920.gq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 199920.gq1 | 199920k8 | \([0, 1, 0, -10427094960, -409519721213100]\) | \(260174968233082037895439009/223081361502731896500\) | \(107500745111285370434881536000\) | \([2]\) | \(254803968\) | \(4.4973\) | |
| 199920.gq2 | 199920k7 | \([0, 1, 0, -6848134960, 215811600674900]\) | \(73704237235978088924479009/899277423164136103500\) | \(433353070828902188812990464000\) | \([4]\) | \(254803968\) | \(4.4973\) | |
| 199920.gq3 | 199920k4 | \([0, 1, 0, -6827954800, 217159842583508]\) | \(73054578035931991395831649/136386452160\) | \(65723308892863856640\) | \([4]\) | \(84934656\) | \(3.9480\) | |
| 199920.gq4 | 199920k6 | \([0, 1, 0, -797614960, -3321292269100]\) | \(116454264690812369959009/57505157319440250000\) | \(27711177742232887182336000000\) | \([2, 2]\) | \(127401984\) | \(4.1507\) | |
| 199920.gq5 | 199920k5 | \([0, 1, 0, -448076400, 3035086226388]\) | \(20645800966247918737249/3688936444974392640\) | \(1777662704905389341504962560\) | \([2]\) | \(84934656\) | \(3.9480\) | |
| 199920.gq6 | 199920k2 | \([0, 1, 0, -426751600, 3392941960148]\) | \(17836145204788591940449/770635366502400\) | \(371361711036992952729600\) | \([2, 2]\) | \(42467328\) | \(3.6014\) | |
| 199920.gq7 | 199920k1 | \([0, 1, 0, -25343600, 58525985748]\) | \(-3735772816268612449/909650165760000\) | \(-438351594911736791040000\) | \([2]\) | \(21233664\) | \(3.2549\) | \(\Gamma_0(N)\)-optimal |
| 199920.gq8 | 199920k3 | \([0, 1, 0, 182385040, -398148269100]\) | \(1392333139184610040991/947901937500000000\) | \(-456784752824064000000000000\) | \([2]\) | \(63700992\) | \(3.8042\) |
Rank
sage: E.rank()
The elliptic curves in class 199920.gq have rank \(0\).
Complex multiplication
The elliptic curves in class 199920.gq do not have complex multiplication.Modular form 199920.2.a.gq
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
