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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 162624.cg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.cg1 | 162624ia4 | \([0, -1, 0, -109446113, 440711201889]\) | \(312196988566716625/25367712678\) | \(11780869440025489047552\) | \([2]\) | \(13271040\) | \(3.2802\) | |
| 162624.cg2 | 162624ia3 | \([0, -1, 0, -6373473, 7867957473]\) | \(-61653281712625/21875235228\) | \(-10158948639244526026752\) | \([2]\) | \(6635520\) | \(2.9337\) | |
| 162624.cg3 | 162624ia2 | \([0, -1, 0, -2811233, -908999199]\) | \(5290763640625/2291573592\) | \(1064216141291090608128\) | \([2]\) | \(4423680\) | \(2.7309\) | |
| 162624.cg4 | 162624ia1 | \([0, -1, 0, 596127, -105543711]\) | \(50447927375/39517632\) | \(-18352149800826175488\) | \([2]\) | \(2211840\) | \(2.3843\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162624.cg have rank \(0\).
Complex multiplication
The elliptic curves in class 162624.cg do not have complex multiplication.Modular form 162624.2.a.cg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
