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SageMath
E = EllipticCurve("hg1")
E.isogeny_class()
Elliptic curves in class 73920hg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.gn6 | 73920hg1 | \([0, 1, 0, -18338145, -30232150497]\) | \(2601656892010848045529/56330588160\) | \(14766725702615040\) | \([2]\) | \(2654208\) | \(2.6295\) | \(\Gamma_0(N)\)-optimal |
| 73920.gn5 | 73920hg2 | \([0, 1, 0, -18358625, -30161261025]\) | \(2610383204210122997209/12104550027662400\) | \(3173135162451532185600\) | \([2, 2]\) | \(5308416\) | \(2.9761\) | |
| 73920.gn4 | 73920hg3 | \([0, 1, 0, -19567905, -25947461025]\) | \(3160944030998056790089/720291785342976000\) | \(188820169776949100544000\) | \([2]\) | \(7962624\) | \(3.1788\) | |
| 73920.gn7 | 73920hg4 | \([0, 1, 0, -9027425, -60788125665]\) | \(-310366976336070130009/5909282337130963560\) | \(-1549082908984859311472640\) | \([2]\) | \(10616832\) | \(3.3227\) | |
| 73920.gn3 | 73920hg5 | \([0, 1, 0, -28017505, 5002857503]\) | \(9278380528613437145689/5328033205714065000\) | \(1396711936678707855360000\) | \([4]\) | \(10616832\) | \(3.3227\) | |
| 73920.gn2 | 73920hg6 | \([0, 1, 0, -103453985, 382795852383]\) | \(467116778179943012100169/28800309694464000000\) | \(7549828384545570816000000\) | \([2, 2]\) | \(15925248\) | \(3.5254\) | |
| 73920.gn8 | 73920hg7 | \([0, 1, 0, 80866015, 1598607436383]\) | \(223090928422700449019831/4340371122724101696000\) | \(-1137802247595386914996224000\) | \([2]\) | \(31850496\) | \(3.8720\) | |
| 73920.gn1 | 73920hg8 | \([0, 1, 0, -1629951265, 25327898503775]\) | \(1826870018430810435423307849/7641104625000000000\) | \(2003069730816000000000000\) | \([4]\) | \(31850496\) | \(3.8720\) |
Rank
sage: E.rank()
The elliptic curves in class 73920hg have rank \(0\).
Complex multiplication
The elliptic curves in class 73920hg do not have complex multiplication.Modular form 73920.2.a.hg
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
