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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 54720fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54720.ct4 | 54720fc1 | \([0, 0, 0, -1094412, 460572176]\) | \(-758575480593601/40535043840\) | \(-7746367510114467840\) | \([2]\) | \(1474560\) | \(2.3834\) | \(\Gamma_0(N)\)-optimal |
| 54720.ct3 | 54720fc2 | \([0, 0, 0, -17729292, 28733214224]\) | \(3225005357698077121/8526675600\) | \(1629473082546585600\) | \([2, 2]\) | \(2949120\) | \(2.7299\) | |
| 54720.ct2 | 54720fc3 | \([0, 0, 0, -17948172, 27987358736]\) | \(3345930611358906241/165622259047500\) | \(31650906595820175360000\) | \([2]\) | \(5898240\) | \(3.0765\) | |
| 54720.ct1 | 54720fc4 | \([0, 0, 0, -283668492, 1838928160784]\) | \(13209596798923694545921/92340\) | \(17646448803840\) | \([2]\) | \(5898240\) | \(3.0765\) |
Rank
sage: E.rank()
The elliptic curves in class 54720fc have rank \(1\).
Complex multiplication
The elliptic curves in class 54720fc do not have complex multiplication.Modular form 54720.2.a.fc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
