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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 6930.bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.bl1 | 6930bl7 | \([1, -1, 1, -229211897, 1335736852121]\) | \(1826870018430810435423307849/7641104625000000000\) | \(5570365271625000000000\) | \([6]\) | \(1327104\) | \(3.3815\) | |
| 6930.bl2 | 6930bl6 | \([1, -1, 1, -14548217, 20191955609]\) | \(467116778179943012100169/28800309694464000000\) | \(20995425767264256000000\) | \([2, 6]\) | \(663552\) | \(3.0350\) | |
| 6930.bl3 | 6930bl4 | \([1, -1, 1, -3939962, 265300049]\) | \(9278380528613437145689/5328033205714065000\) | \(3884136206965553385000\) | \([2]\) | \(442368\) | \(2.8322\) | |
| 6930.bl4 | 6930bl3 | \([1, -1, 1, -2751737, -1367291239]\) | \(3160944030998056790089/720291785342976000\) | \(525092711515029504000\) | \([6]\) | \(331776\) | \(2.6884\) | |
| 6930.bl5 | 6930bl2 | \([1, -1, 1, -2581682, -1589567119]\) | \(2610383204210122997209/12104550027662400\) | \(8824216970165889600\) | \([2, 2]\) | \(221184\) | \(2.4857\) | |
| 6930.bl6 | 6930bl1 | \([1, -1, 1, -2578802, -1593306511]\) | \(2601656892010848045529/56330588160\) | \(41064998768640\) | \([2]\) | \(110592\) | \(2.1391\) | \(\Gamma_0(N)\)-optimal |
| 6930.bl7 | 6930bl5 | \([1, -1, 1, -1269482, -3205147759]\) | \(-310366976336070130009/5909282337130963560\) | \(-4307866823768472435240\) | \([2]\) | \(442368\) | \(2.8322\) | |
| 6930.bl8 | 6930bl8 | \([1, -1, 1, 11371783, 84297299609]\) | \(223090928422700449019831/4340371122724101696000\) | \(-3164130548465870136384000\) | \([6]\) | \(1327104\) | \(3.3815\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.bl have rank \(0\).
Complex multiplication
The elliptic curves in class 6930.bl do not have complex multiplication.Modular form 6930.2.a.bl
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.
