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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 253575.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 253575.z1 | 253575z3 | \([1, -1, 1, -55687505, 159961543872]\) | \(14251520160844849/264449745\) | \(354387950438892890625\) | \([2]\) | \(17694720\) | \(3.0680\) | |
| 253575.z2 | 253575z2 | \([1, -1, 1, -3594380, 2327747622]\) | \(3832302404449/472410225\) | \(633074883109175390625\) | \([2, 2]\) | \(8847360\) | \(2.7214\) | |
| 253575.z3 | 253575z1 | \([1, -1, 1, -893255, -286941378]\) | \(58818484369/7455105\) | \(9990553710901640625\) | \([2]\) | \(4423680\) | \(2.3749\) | \(\Gamma_0(N)\)-optimal |
| 253575.z4 | 253575z4 | \([1, -1, 1, 5280745, 12001633872]\) | \(12152722588271/53476250625\) | \(-71663290339532431640625\) | \([2]\) | \(17694720\) | \(3.0680\) |
Rank
sage: E.rank()
The elliptic curves in class 253575.z have rank \(1\).
Complex multiplication
The elliptic curves in class 253575.z do not have complex multiplication.Modular form 253575.2.a.z
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.
