# Condition of GRH on a sage program

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### Subramani Muthukrishnan

Jan 12, 2021, 5:26:55 AM1/12/21
to sage-nt
Dear All,

I am M. Subramani, working at the Indian Institute of Information Technology D&M, Chennai. I would like to know whether the following sage code uses GRH or not. Kindly looking forward to your suggestions/comments.

sage: x = polygen(QQ)
sage: K.<a,b> = NumberField([x^2+67,x^2+163])
sage: K.absolute_discriminant()
sage: UK = UnitGroup(K); UK
sage: UK.gens_values()
sage: f = K.factor(47); f
sage: [i.residue_class_degree() for i, _ in f]
sage: g = K.factor(167); g
sage: [i.residue_class_degree() for i, _ in g]
sage: I = K.ideal((102005618195957366438*b - 670427217454663892839)*a + 859657055484696603731/2*b + 21319906288123697979267/2)^2
sage: I.small_residue(-113532749239375013512100611401835237324223211*a + 72788832531850093592405531048689006688627546*b)^(71*35)

with regards,
M. Subramani

### John Cremona

Jan 12, 2021, 5:47:25 AM1/12/21
to sage-nt
Although Sage uses the pari library for number field computations, and
by default the pari library does make assumptions such as the GRH, for
speed, Sage's philosophy is that computations should not make any
silent assumptions, so that the results obtained from the pari library
are certified by Sage. Hence, unless you explicitly ask Sage to
assume GRH -- which is often useful as it can speed up computations
considerably -- the output will be unconditional.

John Cremona
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