Exploring the Galois Universe

A new exceptional pair (28679999,2284521) for d=-8 with lmd=2
(2024, January 6)

Gal-Zeta (mp4)

Parings of p-units in the 4p-cyclotomic field

cup14-64.c + bcn3.txt -> (4.1) relation3x.txt, (4.2) relation3w.txt
cup14x-64.c + bcn4.txt -> (4.3) relation4.txt
analnew1.c + relation**all.txt -> zero**.txt + irreg**.txt + index**.txt
Relations : (4.1) relation3xall.txt, (4.2) arelation3wall.txt, (4.3) relation4all.txt (each 280MB)
Distribution of the number of zeros: (4.1) zero3x.txt, (4.2) zero3w.txt, (4.3) zero4.txt
Indices of zeros: (4.1) index3x.txt, (4.2) index3w.txt, (4.3) index4.txt

How to Explore

Let's use gcc and gmp!

1 Download
real quadratic(64-bit)bcn64.c img quadratic (64-bit)bcm64.c
real quadratic (32-bit)bcn32.c img quadratic (32-bit)bcm32.c

2 Edit
Change pn1, pn2 and fn as you like. p should be less than 160 million (64-bit cpu).
Prime-(Mem+Swap): 10M-1.5G, 20M-3G, 40M-6G, 80M-12G, 160M-15G.
f0 and 2*deg*gap should be less than 2^31 (32-bit cpu).

3 Compile and Execute
# gcc -O2 -m64 -o bcn64 bcn64.c -L /usr/local/lib/ -lgmp
# limit stacksize unlimited (or ***M)
# nohup ./bcn64 &

# gcc -O2 -o bcn32 bcn32.c -L /usr/local/lib/ -lgmp
# limit stacksize unlimited (or ***M)
# nohup ./bcn32 &


|d|<200, p<1,000,000 (double-checked)
real100 real100x
img100 img100x
bcn1.txt: exceptional pairs
[prime p, index k, conductor f, type]
bcn2.txt: p divides conductor
bcn3.txt: irregular pairs
[prime p, index k, conductor f, (root/prime) mod p, type]
d=1,5,8, p<20,000,000 (I am exploring up to 40,000,000.)
real1 real5 real8
d=-3, -4, -7, -8, p<20,000,000 (I am exploring up to 40,000,000.)
img3 img4 img7 img8
Exceptional (They are dedicated to Euler.)
34301 and 157229
real1e real5e real8e (34301, 157229)
img3e img4e (379) img7e img8e New!
Excel file 1(|d|<200, p<1,000,000)
Excel file 2(|d|<10, p<20,000,000)

Check of exceptional pairs on v_p(a_0), v_p(b_0) and lambda^- by UBASIC

programs and results
programs: IA7SS.UB (real), IA7SSIM.UB (img)
list of exceptional pairs: ia7ss.txt (real), ia7ssim.txt (img)
[prime p, index k, conductor f]
results: IA7SS.UBD (real), IA7SSIM.UBD (img)
[conductor f, prime p, index k
lambda: Coeff(0), Coeff(1), Coeff(2) mod p
v_p(b_0)=e*: Coeff(0) mod p^2, Coeff(0) mod p^3
v_p(a_0)=e: Coeff'(0) mod p^2, Coeff'(0) mod p^3 ]

Check of exceptional pairs on nu^+ by Gauss sums

programs and results
programs: gauss64.c (small), gauss64d.c (large)
results: g-sum1.txt, g-sum2.txt (small), g-sum3.txt, g-sum4.txt (large)
[prime p, conductor f, index k
auxiliary prime l for cyclotomic unit
auxiliary prime l* for gauss sum
ggsum=(gauss sum for l) mod l*
gsum=ggsum^{(l*-1)/p} mod l*]

Euler is the first explorer.
[Imaginary quadratic Gal -200<d<0]
[Real quadratic Gal 0<d<200]

Partially supported by Grant-in-Aid for Young Scientists, (No. 16740019),
the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Partially supported by JSPS KAKENHI Grant Number 21540018, 25400013, 17K05176 and JP20H00115.

Reg: 2005, June
Title: Exploring the Galois Universe
EP: 379