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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 327990.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 327990.s1 | 327990s6 | \([1, 0, 1, -1054105213, -13172755194412]\) | \(217764763259392950709681/191615146362900\) | \(113977157713481249190900\) | \([2]\) | \(137625600\) | \(3.7241\) | |
| 327990.s2 | 327990s4 | \([1, 0, 1, -66350713, -202748405812]\) | \(54309086480107021681/1575939143610000\) | \(937405355095996128810000\) | \([2, 2]\) | \(68812800\) | \(3.3776\) | |
| 327990.s3 | 327990s2 | \([1, 0, 1, -9768233, 7263126956]\) | \(173294065906331761/61964605497600\) | \(36857992426537289529600\) | \([2, 2]\) | \(34406400\) | \(3.0310\) | |
| 327990.s4 | 327990s1 | \([1, 0, 1, -8691753, 9860027308]\) | \(122083727651299441/32242728960\) | \(19178727118090076160\) | \([2]\) | \(17203200\) | \(2.6844\) | \(\Gamma_0(N)\)-optimal |
| 327990.s5 | 327990s5 | \([1, 0, 1, 16084107, -673253384444]\) | \(773618103830753999/329643718157812500\) | \(-196079771181418033345312500\) | \([4]\) | \(137625600\) | \(3.7241\) | |
| 327990.s6 | 327990s3 | \([1, 0, 1, 29590567, 51077343116]\) | \(4817210305461175439/4682306425314960\) | \(-2785145057845482978182160\) | \([2]\) | \(68812800\) | \(3.3776\) |
Rank
sage: E.rank()
The elliptic curves in class 327990.s have rank \(0\).
Complex multiplication
The elliptic curves in class 327990.s do not have complex multiplication.Modular form 327990.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
