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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 5355p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5355.s4 | 5355p1 | \([1, -1, 0, -1629, 57928]\) | \(-656008386769/1581036975\) | \(-1152575954775\) | \([2]\) | \(6144\) | \(1.0025\) | \(\Gamma_0(N)\)-optimal |
| 5355.s3 | 5355p2 | \([1, -1, 0, -34434, 2465815]\) | \(6193921595708449/6452105625\) | \(4703585000625\) | \([2, 2]\) | \(12288\) | \(1.3491\) | |
| 5355.s2 | 5355p3 | \([1, -1, 0, -42939, 1161148]\) | \(12010404962647729/6166198828125\) | \(4495158945703125\) | \([2]\) | \(24576\) | \(1.6957\) | |
| 5355.s1 | 5355p4 | \([1, -1, 0, -550809, 157481590]\) | \(25351269426118370449/27551475\) | \(20085025275\) | \([4]\) | \(24576\) | \(1.6957\) |
Rank
sage: E.rank()
The elliptic curves in class 5355p have rank \(1\).
Complex multiplication
The elliptic curves in class 5355p do not have complex multiplication.Modular form 5355.2.a.p
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.
