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SageMath
E = EllipticCurve("pw1")
E.isogeny_class()
Elliptic curves in class 381150.pw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 381150.pw1 | 381150pw8 | \([1, -1, 1, -693365987480, -222223511647850853]\) | \(1826870018430810435423307849/7641104625000000000\) | \(154191279233832134765625000000000\) | \([2]\) | \(3822059520\) | \(5.3852\) | |
| 381150.pw2 | 381150pw6 | \([1, -1, 1, -44008355480, -3358820497514853]\) | \(467116778179943012100169/28800309694464000000\) | \(581166835432506759744000000000000\) | \([2, 2]\) | \(1911029760\) | \(5.0386\) | |
| 381150.pw3 | 381150pw5 | \([1, -1, 1, -11918384105, -43972438319103]\) | \(9278380528613437145689/5328033205714065000\) | \(107515378483564105004437265625000\) | \([2]\) | \(1274019840\) | \(4.8359\) | |
| 381150.pw4 | 381150pw3 | \([1, -1, 1, -8324003480, 227599615893147]\) | \(3160944030998056790089/720291785342976000\) | \(14534902642254330986496000000000\) | \([2]\) | \(955514880\) | \(4.6921\) | |
| 381150.pw5 | 381150pw2 | \([1, -1, 1, -7809587105, 264573563598897]\) | \(2610383204210122997209/12104550027662400\) | \(244259978748188336651025000000\) | \([2, 2]\) | \(637009920\) | \(4.4893\) | |
| 381150.pw6 | 381150pw1 | \([1, -1, 1, -7800875105, 265195582974897]\) | \(2601656892010848045529/56330588160\) | \(1136705473180791360000000\) | \([2]\) | \(318504960\) | \(4.1428\) | \(\Gamma_0(N)\)-optimal |
| 381150.pw7 | 381150pw4 | \([1, -1, 1, -3840182105, 533310220908897]\) | \(-310366976336070130009/5909282337130963560\) | \(-119244513409095293685097025625000\) | \([2]\) | \(1274019840\) | \(4.8359\) | |
| 381150.pw8 | 381150pw7 | \([1, -1, 1, 34399644520, -14025444817514853]\) | \(223090928422700449019831/4340371122724101696000\) | \(-87585160602667896323165241000000000\) | \([2]\) | \(3822059520\) | \(5.3852\) |
Rank
sage: E.rank()
The elliptic curves in class 381150.pw have rank \(1\).
Complex multiplication
The elliptic curves in class 381150.pw do not have complex multiplication.Modular form 381150.2.a.pw
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.
