IGUSA INVARIANTS

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These programs compute Igusa invariants of sextics and other
associated data; C will be the corresponding curve of genus 2, J the
Jacobian of C, and z the moduli point (J, canonical polarization).
The algorithms are based on classical invariants of sextics as reworked by Mestre.

Double click on highlighted text to get the functions evaluated.

OPTION 1: Type in a polynomial f in x of degree at most 6
(and then double-clik on it):
f=0 Result
and compute its Igusa invariants v=[J2, J4, J6, J8, J10]:
v=jinv(f) Result
OPTION 2: Type in a vector of Igusa invariants [J2, J4, J6, J10]
(and then double-clik on it):

After evaluating we have v=[J2, J4, J6, J8, J10] (i.e. J8, a polynomial in the other invariants, is computed).

v=setinv([0,0,0,0]) Result
Factorization of J10:
smallfact(v[5]) Result
H4 invariant:

(The vanishing of this invariant means that the curve C has an automorphism other than the hyperelliptic involution.
Equivalently, z is in the Humbert surface H4 or J has a 2x2 isogeny to a product of elliptic curves.)

h4=h4norm(v) Result
smallfact(h4) Result
H5 invariant:

(The vanishing of this invariant means that z is in the Humbert surface H5;
i.e. the endomorphism ring of J tensored with Q has a square root of 5).
h5=h5norm(v) Result
smallfact(h5) Result
Matrix invariant:
m=matinv(v) Result
Obstruction for model of C over field of moduli:

(Requires matrix m from previous calculation; m must have rational entries).

This function outputs the rank of m if less than 3 or otherwise a list
of primes for which the corresponding conic has no local points (0
corresponds to the infinite prime). Note that the list should contain
an even number of elements. The list is non-empty if and only if there is no
curve defined over Q with equivalent invariants.

globhilb(m) Result
Stable reduction type of C:

This function will compute the stable reduction type of C ( based on
work of Q. Liu). v must be an integer valued vector of Igusa invariants.

redtp(v) Result
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Last updated Nov 12, 1997.
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