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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 133518u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133518.by4 | 133518u1 | \([1, 1, 1, 22247, 768743]\) | \(50447927375/39517632\) | \(-953859569116608\) | \([2]\) | \(497664\) | \(1.5623\) | \(\Gamma_0(N)\)-optimal |
| 133518.by3 | 133518u2 | \([1, 1, 1, -104913, 6516375]\) | \(5290763640625/2291573592\) | \(55313015695477848\) | \([2]\) | \(995328\) | \(1.9089\) | |
| 133518.by2 | 133518u3 | \([1, 1, 1, -237853, -56806993]\) | \(-61653281712625/21875235228\) | \(-528014999707080732\) | \([2]\) | \(1492992\) | \(2.1116\) | |
| 133518.by1 | 133518u4 | \([1, 1, 1, -4084443, -3178699437]\) | \(312196988566716625/25367712678\) | \(612314915137399782\) | \([2]\) | \(2985984\) | \(2.4582\) |
Rank
sage: E.rank()
The elliptic curves in class 133518u have rank \(1\).
Complex multiplication
The elliptic curves in class 133518u do not have complex multiplication.Modular form 133518.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
