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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 247962.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 247962.bc1 | 247962bc4 | \([1, 0, 0, -3834169, -2871235471]\) | \(258252149810350513/1938176193096\) | \(46782861595012023624\) | \([2]\) | \(11796480\) | \(2.6045\) | |
| 247962.bc2 | 247962bc2 | \([1, 0, 0, -400849, 23053289]\) | \(295102348042033/161237583936\) | \(3891883307648491584\) | \([2, 2]\) | \(5898240\) | \(2.2579\) | |
| 247962.bc3 | 247962bc1 | \([1, 0, 0, -308369, 65797545]\) | \(134351465835313/205590528\) | \(4962455555346432\) | \([2]\) | \(2949120\) | \(1.9113\) | \(\Gamma_0(N)\)-optimal |
| 247962.bc4 | 247962bc3 | \([1, 0, 0, 1552791, 182079585]\) | \(17154149157653327/10519679024712\) | \(-253919478316838605128\) | \([2]\) | \(11796480\) | \(2.6045\) |
Rank
sage: E.rank()
The elliptic curves in class 247962.bc have rank \(2\).
Complex multiplication
The elliptic curves in class 247962.bc do not have complex multiplication.Modular form 247962.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
