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- Title
ON DISCRIMINANTS OF MINIMAL POLYNOMIALS OF THE RAMANUJAN $t_n$ CLASS INVARIANTS.
- Authors
CHAVAN, SARTH
- Abstract
We study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$. We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$ , where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$ , where j is the j -invariant, divides $\Delta (\mathcal {P}_n)$. Moreover, using Ye's computation of $\log|\Delta(H_n)|$ ['Revisiting the Gross–Zagier discriminant formula', Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$ , and thus $\Delta(\mathcal{P}_n)$ , for all squarefree $n\equiv11\pmod{24}$.
- Subjects
POLYNOMIALS; POLYNOMIAL rings; ELLIPTIC curves; MATHEMATICS; GROBNER bases
- Publication
Bulletin of the Australian Mathematical Society, 2023, Vol 108, Issue 2, p264
- ISSN
0004-9727
- Publication type
Article
- DOI
10.1017/S0004972723000278