# Can an elliptic curve have the form y^2 ≡ x^2 + 2x + 2 mod 17?

I'm new to cryptography and the associated level of maths. I'm practising past papers for an exam and found the question:

Show that the condition 4a^3 + 27b^2 ≠ 0 mod p is fulfilled for the elliptic curve y^2 ≡ x^2 + 2x + 2 mod 17.

Surely elliptic curves usually follow the format y^2 ≡ x^3 + ax + b mod p?

The x^2 has thrown me on this - can anyone clarify if this looks like a typo?

Many thanks, Paul

• Surely the definition of an elliptic curve has been given in class. What is it? – fkraiem May 15 '18 at 11:57
• Yes, it was given as: y^2 ≡ x^3 + ax + b mod p – Paul May 15 '18 at 12:00
• You guessed it, what's in the question does not match your definition. My bets are on a typo. – fgrieu May 15 '18 at 12:25
• Yes, I thought that might be the case, but as my mathematical ability is very basic, I did not want to presume anything. Thanks @fgrieu – Paul May 15 '18 at 12:33
• In fact, the question cannot possibly be correct as stated (even if a different curve model had been introduced in class): That curve has genus $0$, hence is (by definition) certainly not an elliptic curve. – yyyyyyy May 15 '18 at 15:11