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I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full in order to find an explicit arithmetic Fuchsian group of a genus 2 surface and its Hilbert symbol...

In the paper, the author gave a table (table 1 and 2) of quaternion algebras with the places where they are ramified (the product of the primes are denoted as $\Delta(A)$ there). More specifically, I hope to find a quaternion algebra coming from the data in the 4th row of table 1, i.e., a quaternion algebra $A=(\frac{a,b}{\mathbb{Q}})$ and $\Delta(A)=\{2\cdot 13\}$... (I am not sure if this is sufficient data.) It looks to me that it is not unique.

This should relate to the following post: Explicit description of a quaternion algebra with a prescribed set of ramified places but in the accepted answer I do not understand how this $73$ and $\infty$ are taken...

I searched my head for the algebra classes that I took so many years ago but I don't think it is sufficient... Any help would be very much appreciated !

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    $\begingroup$ Hi! You should read Voight's book on quaternion algebras. However, if you are only looking for explicit pairs (a,b), then Magma or Pari/GP will compute them for you. For instance in the special case you are asking, (a,b)=(-5,26) will do. $\endgroup$
    – Aurel
    Commented Nov 27 at 15:17

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Check out Exercise 14.9 in my book (http://quatalg.org). You want a quaternion algebra of discriminant $\Delta(A)=D=26$. You can take $b=D=26$, and then you need to find an odd prime $q \equiv 5 \pmod{8}$ coprime to $13$ such that $q$ is a nonsquare modulo $13$, and take $(q,D\,|\,\mathbb{Q})$.

The command in Magma is

> QuaternionAlgebra(26);
Quaternion Algebra with base ring Rational Field, defined by i^2 = 26, j^2 = -5

(Of course, such a representation is not unique.)

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