Your computation asks for *a* square roots of $275$ *modulo* $625$ (since $-84100 = 275\mod 625$). There exist many such square roots (amongst which $30$ and $280$), as shown by the following computation:

```
sage: Zmod(625)(275).sqrt(all=True)
[30, 95, 155, 220, 280, 345, 405, 470, 530, 595]
```

Thus, the functions `sqrt`

in Magma and SageMath both give sensible results. They simply do not make the same choice when they choose one of the possible square roots.

**Note** : There exist two implementations in SageMath for the computation of square roots in $\mathbb Z/n\mathbb Z$. One is specific for small values of $n$ (32 bits) and the other for larger values of $n$. It happens that they do not to make the same choice for a square root. In your case, `sqrt`

is the method for small values of $n$. But one can specifically call the other implementation using `square_root`

.

The method for small values return the smallest (using the natural order on integers) square root, that is $30$ in your case, using a brute force algorithm. The algorithm used for large values is based on Newton method.

**Important remark:** My explanation are specific to the value $625$ which is a prime power, different from $3$ *modulo* $4$.

**Questions for knowledgeable SageMath developers:**

- The alias
`square_root = sqrt`

is defined in `IntegerMod_abstract`

but not in `IntegerMod_int`

, whence the different answers. Should it be defined also in `IntegerMod_int`

for consistency? - Is there a satisfactory way to define a
*preferred* square root in $\mathbb Z/625\mathbb Z$, to ensure consistency between implementations, and to satisfy the *principle of least surprise*?

[EDIT: read second comment first!]There is an inconsistency I do not understand yet: The function`sqrt`

you use simply calls the method`sqrt`

on the value`-Z(D)`

which is an element of`Zmod(625)`

(that is, what you type is equivalent to`Zmod(625)(-84100).sqrt()`

). Actually, elements of`Zmod(625)`

have not only the method`sqrt`

but also a method`square_root`

. It is supposed to be the same thing (`square_root`

is defined as an alias for`sqrt`

), but the two methods do not return the same result!I do not know where this can come!

Both answers make sense, since an element can have distinct square roots. And both $30^2$ and $280^2$ are equal to $-84100$

modulo$625$. The problem then reduces to how to definetheorasquare root in $\mathbb Z/625\mathbb Z$. You can obtain all the square roots using`Zmod(625)(-84100).sqrt(all = True)`

(or replacing with`square_root`

, this time one obtains the same result) :`[30, 95, 155, 220, 280, 345, 405, 470, 530, 595]`

.The inconsistency I mentioned in my first comment concerns the difference between

`sqrt`

and`square_root`

in SageMath.Quite interesting. Using

`square_root`

actually produced the result that I want. Thank you. If you can provide it as an answer, I can accept.OK, I found the origin of the inconsistency, I can write an answer!