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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 133518.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133518.z1 | 133518bl3 | \([1, 0, 1, -26589307, 52770396134]\) | \(86129359107301290313/9166294368\) | \(221252062781911392\) | \([2]\) | \(8847360\) | \(2.7559\) | |
| 133518.z2 | 133518bl2 | \([1, 0, 1, -1665947, 820144550]\) | \(21184262604460873/216872764416\) | \(5234781315311944704\) | \([2, 2]\) | \(4423680\) | \(2.4093\) | |
| 133518.z3 | 133518bl4 | \([1, 0, 1, -417467, 2021182310]\) | \(-333345918055753/72923718045024\) | \(-1760201276048311906656\) | \([2]\) | \(8847360\) | \(2.7559\) | |
| 133518.z4 | 133518bl1 | \([1, 0, 1, -186267, -10251866]\) | \(29609739866953/15259926528\) | \(368337529504530432\) | \([2]\) | \(2211840\) | \(2.0628\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 133518.z have rank \(1\).
Complex multiplication
The elliptic curves in class 133518.z do not have complex multiplication.Modular form 133518.2.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.
