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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 55545.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55545.c1 | 55545f4 | \([1, 1, 1, -2671990, -1682212750]\) | \(14251520160844849/264449745\) | \(39148053096898305\) | \([2]\) | \(1013760\) | \(2.3088\) | |
| 55545.c2 | 55545f2 | \([1, 1, 1, -172465, -24527770]\) | \(3832302404449/472410225\) | \(69933667630565025\) | \([2, 2]\) | \(506880\) | \(1.9622\) | |
| 55545.c3 | 55545f1 | \([1, 1, 1, -42860, 3000332]\) | \(58818484369/7455105\) | \(1103623096263345\) | \([4]\) | \(253440\) | \(1.6156\) | \(\Gamma_0(N)\)-optimal |
| 55545.c4 | 55545f3 | \([1, 1, 1, 253380, -126049218]\) | \(12152722588271/53476250625\) | \(-7916404301658680625\) | \([2]\) | \(1013760\) | \(2.3088\) |
Rank
sage: E.rank()
The elliptic curves in class 55545.c have rank \(0\).
Complex multiplication
The elliptic curves in class 55545.c do not have complex multiplication.Modular form 55545.2.a.c
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.
