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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 192027.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 192027.q1 | 192027n6 | \([1, 0, 1, -5959495270, -177077568797197]\) | \(89254274298475942657/17457\) | \(4578178684047990153\) | \([2]\) | \(64880640\) | \(3.8818\) | |
| 192027.q2 | 192027n4 | \([1, 0, 1, -372469705, -2766840789649]\) | \(21790813729717297/304746849\) | \(79921265287425764100921\) | \([2, 2]\) | \(32440320\) | \(3.5352\) | |
| 192027.q3 | 192027n5 | \([1, 0, 1, -361908220, -2931126801121]\) | \(-19989223566735457/2584262514273\) | \(-677734751493894340962601017\) | \([2]\) | \(64880640\) | \(3.8818\) | |
| 192027.q4 | 192027n3 | \([1, 0, 1, -90190015, 285909210959]\) | \(309368403125137/44372288367\) | \(11636837071904107345803543\) | \([2]\) | \(32440320\) | \(3.5352\) | |
| 192027.q5 | 192027n2 | \([1, 0, 1, -23940700, -40646912539]\) | \(5786435182177/627352209\) | \(164526007368632622128361\) | \([2, 2]\) | \(16220160\) | \(3.1887\) | |
| 192027.q6 | 192027n1 | \([1, 0, 1, 1982945, -3150952411]\) | \(3288008303/18259263\) | \(-4788575852267065178727\) | \([2]\) | \(8110080\) | \(2.8421\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 192027.q have rank \(0\).
Complex multiplication
The elliptic curves in class 192027.q do not have complex multiplication.Modular form 192027.2.a.q
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
