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ON DISCRIMINANTS OF MINIMAL POLYNOMIALS OF THE RAMANUJAN
$t_n$ CLASS INVARIANTS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
- Published online by Cambridge University Press:
- 11 April 2023, pp. 264-275
- Print publication:
- October 2023
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- Article
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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 
$n\equiv11\pmod{24}$.
