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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 390390ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390390.ep6 | 390390ep1 | \([1, 0, 0, -48424165, 129696365057]\) | \(2601656892010848045529/56330588160\) | \(271896989905981440\) | \([2]\) | \(31850496\) | \(2.8723\) | \(\Gamma_0(N)\)-optimal |
| 390390.ep5 | 390390ep2 | \([1, 0, 0, -48478245, 129392143425]\) | \(2610383204210122997209/12104550027662400\) | \(58426351014471121281600\) | \([2, 2]\) | \(63700992\) | \(3.2188\) | |
| 390390.ep4 | 390390ep3 | \([1, 0, 0, -51671500, 111308666000]\) | \(3160944030998056790089/720291785342976000\) | \(3476710872119544643584000\) | \([2]\) | \(95551488\) | \(3.4216\) | |
| 390390.ep7 | 390390ep4 | \([1, 0, 0, -23838045, 260827898265]\) | \(-310366976336070130009/5909282337130963560\) | \(-28522977168404769090080040\) | \([2]\) | \(127401984\) | \(3.5654\) | |
| 390390.ep3 | 390390ep5 | \([1, 0, 0, -73983725, -21513579543]\) | \(9278380528613437145689/5328033205714065000\) | \(25717398629639500368585000\) | \([2]\) | \(127401984\) | \(3.5654\) | |
| 390390.ep2 | 390390ep6 | \([1, 0, 0, -273183180, -1642753723248]\) | \(467116778179943012100169/28800309694464000000\) | \(139013594036026085376000000\) | \([2, 2]\) | \(191102976\) | \(3.7681\) | |
| 390390.ep8 | 390390ep7 | \([1, 0, 0, 213536820, -6859516027248]\) | \(223090928422700449019831/4340371122724101696000\) | \(-20950142398504798583168064000\) | \([2]\) | \(382205952\) | \(4.1147\) | |
| 390390.ep1 | 390390ep8 | \([1, 0, 0, -4304090060, -108685098284400]\) | \(1826870018430810435423307849/7641104625000000000\) | \(36882152573891625000000000\) | \([2]\) | \(382205952\) | \(4.1147\) |
Rank
sage: E.rank()
The elliptic curves in class 390390ep have rank \(1\).
Complex multiplication
The elliptic curves in class 390390ep do not have complex multiplication.Modular form 390390.2.a.ep
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.
