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SageMath
E = EllipticCurve("fx1")
E.isogeny_class()
Elliptic curves in class 227136.fx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227136.fx1 | 227136r4 | \([0, 1, 0, -21161729, -37468729089]\) | \(828279937799497/193444524\) | \(244769035241105915904\) | \([2]\) | \(12386304\) | \(2.9028\) | |
| 227136.fx2 | 227136r2 | \([0, 1, 0, -1476609, -441018369]\) | \(281397674377/96589584\) | \(122216637623816945664\) | \([2, 2]\) | \(6193152\) | \(2.5563\) | |
| 227136.fx3 | 227136r1 | \([0, 1, 0, -611329, 178695167]\) | \(19968681097/628992\) | \(795875540081836032\) | \([2]\) | \(3096576\) | \(2.2097\) | \(\Gamma_0(N)\)-optimal |
| 227136.fx4 | 227136r3 | \([0, 1, 0, 4364031, -3061129473]\) | \(7264187703863/7406095788\) | \(-9371073853359519694848\) | \([2]\) | \(12386304\) | \(2.9028\) |
Rank
sage: E.rank()
The elliptic curves in class 227136.fx have rank \(1\).
Complex multiplication
The elliptic curves in class 227136.fx do not have complex multiplication.Modular form 227136.2.a.fx
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.
