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SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 166464.di
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 166464.di1 | 166464fd3 | \([0, 0, 0, -124934700, 406363348528]\) | \(46753267515625/11591221248\) | \(53467536436723553572749312\) | \([2]\) | \(31850496\) | \(3.6471\) | |
| 166464.di2 | 166464fd1 | \([0, 0, 0, -42535020, -106734964304]\) | \(1845026709625/793152\) | \(3658620826272072204288\) | \([2]\) | \(10616832\) | \(3.0978\) | \(\Gamma_0(N)\)-optimal |
| 166464.di3 | 166464fd2 | \([0, 0, 0, -35876460, -141282237008]\) | \(-1107111813625/1228691592\) | \(-5667660987498723853467648\) | \([2]\) | \(21233664\) | \(3.4443\) | |
| 166464.di4 | 166464fd4 | \([0, 0, 0, 301213140, 2576819527216]\) | \(655215969476375/1001033261568\) | \(-4617527458247276110012219392\) | \([2]\) | \(63700992\) | \(3.9936\) |
Rank
sage: E.rank()
The elliptic curves in class 166464.di have rank \(0\).
Complex multiplication
The elliptic curves in class 166464.di do not have complex multiplication.Modular form 166464.2.a.di
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
