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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 34650s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.n6 | 34650s1 | \([1, -1, 0, -64470042, -199227783884]\) | \(2601656892010848045529/56330588160\) | \(641640605760000000\) | \([2]\) | \(2654208\) | \(2.9438\) | \(\Gamma_0(N)\)-optimal |
| 34650.n5 | 34650s2 | \([1, -1, 0, -64542042, -198760431884]\) | \(2610383204210122997209/12104550027662400\) | \(137878390158842025000000\) | \([2, 2]\) | \(5308416\) | \(3.2904\) | |
| 34650.n4 | 34650s3 | \([1, -1, 0, -68793417, -170980198259]\) | \(3160944030998056790089/720291785342976000\) | \(8204573617422336000000000\) | \([2]\) | \(7962624\) | \(3.4931\) | |
| 34650.n7 | 34650s4 | \([1, -1, 0, -31737042, -400675206884]\) | \(-310366976336070130009/5909282337130963560\) | \(-67310419121382381800625000\) | \([2]\) | \(10616832\) | \(3.6370\) | |
| 34650.n3 | 34650s5 | \([1, -1, 0, -98499042, 33064007116]\) | \(9278380528613437145689/5328033205714065000\) | \(60689628233836771640625000\) | \([2]\) | \(10616832\) | \(3.6370\) | |
| 34650.n2 | 34650s6 | \([1, -1, 0, -363705417, 2523630745741]\) | \(467116778179943012100169/28800309694464000000\) | \(328053527613504000000000000\) | \([2, 2]\) | \(15925248\) | \(3.8397\) | |
| 34650.n8 | 34650s7 | \([1, -1, 0, 284294583, 10537446745741]\) | \(223090928422700449019831/4340371122724101696000\) | \(-49439539819779220881000000000\) | \([2]\) | \(31850496\) | \(4.1863\) | |
| 34650.n1 | 34650s8 | \([1, -1, 0, -5730297417, 166961376217741]\) | \(1826870018430810435423307849/7641104625000000000\) | \(87036957369140625000000000\) | \([2]\) | \(31850496\) | \(4.1863\) |
Rank
sage: E.rank()
The elliptic curves in class 34650s have rank \(1\).
Complex multiplication
The elliptic curves in class 34650s do not have complex multiplication.Modular form 34650.2.a.s
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.
