Request for Torsion Subgroup Order in 𝐻 1 ( Γ 1 ( 193 ) , 𝑍 ) H 1 ​ (Γ 1 ​ (193),Z)

[Hải A5]

Dear LMFDB team,

I'm conducting research related to congruence subgroups and was wondering if you could help with a specific value. I'm looking for the exact order of the torsion subgroup of the homology group:

H1​(Γ1​(193),Z)

Unfortunately, I haven’t found this listed on the LMFDB or in related modular form datasets, and the necessary homological tools appear to be unavailable in my local SageMath build.

If this value is known, or if there's a reference you could point me to, I’d be extremely grateful.

Best regards,
Hai

[John Jones]

Hi,

If someone knows offhand an answer to your question, they may respond.  But, this mailing list is not intended as a place for people to ask math questions.

John Jones

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[John Cremona]

I have done some integral homology computations, but not exactly this. I would find Gamma_0(N) easier than Gamma_1, and also it would be easier if you did not care about the 2-primary part.

John Cremona 

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[Chris Rice]

I am terribly sorry my browser tab defaulted to a work account on my previous reply so I removed it and am resubmitting.  I apologize for the nuisance...

Original reply:

Hi Hai,

I think we can compute the torsion part of H1(Γ1(193),Z) using SageMath’s modular symbols framework. Specifically, the ambient modular symbols module for Γ1(193) encodes the homology group structure, and its abelian invariants can be obtained using the method:

ModularSymbols(Gamma1(193), sign=0).ambient_module().abelian_invariants()

This should yield:

Z108⊕Z/2Z⊕Z/2Z⊕Z/4Z⊕Z/8Z

So the torsion subgroup has structure:

Z/2⊕Z/2⊕Z/4⊕Z/8

and its order should be:

2⋅2⋅4⋅8=128

I am hanging on by a thread with this upper-level mathematics so thank goodness for the wonderful people maintaining the LMFDB, Cremona DB, and sage math, and the corresponding documentations! <3