Does that mean that by accident there are almost as many points on the curve as the modulo to compute it?

It is not an accident.

The number of points on an elliptic curve over a finite field is always close to the size of that field, by Hasse's theorem. Specifically it says that the difference between the number of points on the curve (excluding the point at infinity) and the size of the field is at most twice the square root of the field size.

Or are these the same numbers and one is wrong?

No, they are distinct.

The numbers can be computed using the following Sage code:

>>> F = GF(2**256 - 2**32 - 977) # secp256k1 field
>>> E = EllipticCurve(F, [0,7])  # secp256k1 curve
>>> F.order() - (E.order() - 1)
432420386565659656852420866390673177327
>>> int(sqrt(F.order())
340282366920938463463374607431768211455

Does that mean that by accident there are almost as many points on the curve as the modulo to compute it?

It is not an accident.

The number of points on an elliptic curve over a finite field is always close to the size of that field, by Hasse's theorem. Specifically it says that the difference between the number of points on the curve (excluding the point at infinity) and the size of the field is at most twice the square root of the field size.

Or are these the same numbers and one is wrong?

No, they are distinct.

The numbers can be computed using the following Sage code:

>>> F = GF(2**256 - 2**32 - 977) # secp256k1 field
>>> E = EllipticCurve(F, [0,7])  # secp256k1 curve
>>> F.order() + 1 - E.order()
432420386565659656852420866390673177327
>>> int(sqrt(F.order())
340282366920938463463374607431768211455

Does that mean that by accident there are almost as many points on the curve as the modulo to compute it?

It is not an accident.

The number of points on an elliptic curve over a finite field is always close to the size of that field, by Hasse's theorem. Specifically it says that the difference between the number of points on the curve (excluding the point at infinity) and the size of the field is at most twice the square root of the field size.

Or are these the same numbers and one is wrong?

No, they are distinct.

Does that mean that by accident there are almost as many points on the curve as the modulo to compute it?

It is not an accident.

The number of points on an elliptic curve over a finite field is always close to the size of that field, by Hasse's theorem. Specifically it says that the difference between the number of points on the curve (excluding the point at infinity) and the size of the field is at most twice the square root of the field size.

Or are these the same numbers and one is wrong?

No, they are distinct.

The numbers can be computed using the following Sage code:

>>> F = GF(2**256 - 2**32 - 977) # secp256k1 field
>>> E = EllipticCurve(F, [0,7])  # secp256k1 curve
>>> F.order() - (E.order() - 1)
432420386565659656852420866390673177327
>>> int(sqrt(F.order())
340282366920938463463374607431768211455