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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 55440dx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.df6 | 55440dx1 | \([0, 0, 0, -41260827, 102012877514]\) | \(2601656892010848045529/56330588160\) | \(168202234956349440\) | \([2]\) | \(2654208\) | \(2.8322\) | \(\Gamma_0(N)\)-optimal |
| 55440.df5 | 55440dx2 | \([0, 0, 0, -41306907, 101773602506]\) | \(2610383204210122997209/12104550027662400\) | \(36143992709799483801600\) | \([2, 2]\) | \(5308416\) | \(3.1788\) | |
| 55440.df4 | 55440dx3 | \([0, 0, 0, -44027787, 87550667066]\) | \(3160944030998056790089/720291785342976000\) | \(2150779746365560848384000\) | \([2]\) | \(7962624\) | \(3.3815\) | |
| 55440.df7 | 55440dx4 | \([0, 0, 0, -20311707, 205149768266]\) | \(-310366976336070130009/5909282337130963560\) | \(-17645022510155663094743040\) | \([2]\) | \(10616832\) | \(3.5254\) | |
| 55440.df3 | 55440dx5 | \([0, 0, 0, -63039387, -16916163766]\) | \(9278380528613437145689/5328033205714065000\) | \(15909421903730906664960000\) | \([2]\) | \(10616832\) | \(3.5254\) | |
| 55440.df2 | 55440dx6 | \([0, 0, 0, -232771467, -1292052387526]\) | \(467116778179943012100169/28800309694464000000\) | \(85997263942714392576000000\) | \([2, 2]\) | \(15925248\) | \(3.7281\) | |
| 55440.df8 | 55440dx7 | \([0, 0, 0, 181948533, -5395209123526]\) | \(223090928422700449019831/4340371122724101696000\) | \(-12960278726516204078628864000\) | \([2]\) | \(31850496\) | \(4.0747\) | |
| 55440.df1 | 55440dx8 | \([0, 0, 0, -3667390347, -85483491145414]\) | \(1826870018430810435423307849/7641104625000000000\) | \(22816216152576000000000000\) | \([2]\) | \(31850496\) | \(4.0747\) |
Rank
sage: E.rank()
The elliptic curves in class 55440dx have rank \(0\).
Complex multiplication
The elliptic curves in class 55440dx do not have complex multiplication.Modular form 55440.2.a.dx
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.
