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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 127050.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.bt1 | 127050be8 | \([1, 1, 0, -77040665275, 8230474751180125]\) | \(1826870018430810435423307849/7641104625000000000\) | \(211510671102650390625000000000\) | \([2]\) | \(477757440\) | \(4.8359\) | |
| 127050.bt2 | 127050be6 | \([1, 1, 0, -4889817275, 124399129228125]\) | \(467116778179943012100169/28800309694464000000\) | \(797211022541161536000000000000\) | \([2, 2]\) | \(238878720\) | \(4.4893\) | |
| 127050.bt3 | 127050be5 | \([1, 1, 0, -1324264900, 1628167405000]\) | \(9278380528613437145689/5328033205714065000\) | \(147483372405437729772890625000\) | \([2]\) | \(159252480\) | \(4.2866\) | |
| 127050.bt4 | 127050be3 | \([1, 1, 0, -924889275, -8429923699875]\) | \(3160944030998056790089/720291785342976000\) | \(19938138055218561024000000000\) | \([2]\) | \(119439360\) | \(4.1428\) | |
| 127050.bt5 | 127050be2 | \([1, 1, 0, -867731900, -9799310118000]\) | \(2610383204210122997209/12104550027662400\) | \(335061699243056703225000000\) | \([2, 2]\) | \(79626240\) | \(3.9400\) | |
| 127050.bt6 | 127050be1 | \([1, 1, 0, -866763900, -9822347550000]\) | \(2601656892010848045529/56330588160\) | \(1559266767051840000000\) | \([2]\) | \(39813120\) | \(3.5935\) | \(\Gamma_0(N)\)-optimal |
| 127050.bt7 | 127050be4 | \([1, 1, 0, -426686900, -19752372633000]\) | \(-310366976336070130009/5909282337130963560\) | \(-163572720725782295864330625000\) | \([2]\) | \(159252480\) | \(4.2866\) | |
| 127050.bt8 | 127050be7 | \([1, 1, 0, 3822182725, 519462193228125]\) | \(223090928422700449019831/4340371122724101696000\) | \(-120144253227253630072929000000000\) | \([2]\) | \(477757440\) | \(4.8359\) |
Rank
sage: E.rank()
The elliptic curves in class 127050.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 127050.bt do not have complex multiplication.Modular form 127050.2.a.bt
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
