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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 20184f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20184.i5 | 20184f1 | \([0, 1, 0, 561, 6510]\) | \(2048/3\) | \(-28551519408\) | \([2]\) | \(12096\) | \(0.69172\) | \(\Gamma_0(N)\)-optimal |
| 20184.i4 | 20184f2 | \([0, 1, 0, -3644, 62016]\) | \(35152/9\) | \(1370472931584\) | \([2, 2]\) | \(24192\) | \(1.0383\) | |
| 20184.i3 | 20184f3 | \([0, 1, 0, -20464, -1081744]\) | \(1556068/81\) | \(49337025537024\) | \([2, 2]\) | \(48384\) | \(1.3849\) | |
| 20184.i2 | 20184f4 | \([0, 1, 0, -54104, 4825440]\) | \(28756228/3\) | \(1827297242112\) | \([2]\) | \(48384\) | \(1.3849\) | |
| 20184.i1 | 20184f5 | \([0, 1, 0, -323224, -70837648]\) | \(3065617154/9\) | \(10963783452672\) | \([2]\) | \(96768\) | \(1.7314\) | |
| 20184.i6 | 20184f6 | \([0, 1, 0, 13176, -4257360]\) | \(207646/6561\) | \(-7992598136997888\) | \([2]\) | \(96768\) | \(1.7314\) |
Rank
sage: E.rank()
The elliptic curves in class 20184f have rank \(0\).
Complex multiplication
The elliptic curves in class 20184f do not have complex multiplication.Modular form 20184.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
