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In this work, we investigate hyperelliptic curves of type C : y 2 = x 2g+1 + ax g+1 + bx over the finite field Fq, q = p n , p > 2. For the case of g = 3 and 4 we propose algorithms to compute the number of points on the Jacobian of the curve with complexityÕ(log 4 p) andÕ(log 8 p). For curves of genus 2 − 7 we give a complete list of the characteristic polynomials of Frobenius endomorphism modulo p. * The reported study was funded by RFBR according to the research project 18-31-00244. 1The polynomial L X,q (T ) has a formwhere a i ∈ Z and a i ≤ 2g i q i 2 . Coefficients of the polynomial L X,q k (T ) are denoted by a i,k .Let J X (F q ) be the Jacobian of the curve over finite field F q and let χ X,q (T ) be the characteristic polynomial of the Frobenius endomorphism on J X . Then we have L X (T ) = T 2g χ X,q ( 1 T ) and #J X (F q ) = L X,q (1) = χ X,q (1). Because of that, point-counting on the Jacobian is equivalent to determining of χ X,q (T ) (or L X,q (T )).The order of the Jacobian satisfies Hasse-Weil bounds

In this work, we list all possible characteristic polynomials of the Frobenius endomorphism for genus 3 hyperelliptic curves of type y 2 = x 7 + ax 4 + bx over finite field F q of characteristic p > 3.

We present an algorithm for computing the Stickelberger ideal for multiquadratic fields K = Q(√d1,√d2, . . . , √dn), where the integers di ≡ 1 mod 4 for i ∈ {1, . . . , n} or dj ≡ 2 mod 8 for one j ∈ {1, . . . , n}; all di’s are pairwise co-prime and squarefree. Our result is based on the paper of Kuˇcera [J. Number Theory, no. 56, 1996]. The algorithm we present works in time O(lg ∆K • 2n• poly(n)), where ∆K is the discriminant of K. As an interesting application, we show a connection between Stickelberger ideal and the class number of a multiquadratic field

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