sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
gp: K = bnfinit(y^18 - 6*y^17 + 15*y^16 - 12*y^15 - 39*y^14 + 153*y^13 - 265*y^12 + 243*y^11 + 9*y^10 - 417*y^9 + 768*y^8 - 906*y^7 + 751*y^6 - 372*y^5 + 66*y^4 + 19*y^3 - 9*y^2 + 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
x18−6x17+15x16−12x15−39x14+153x13−265x12+243x11+9x10+⋯−1
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree: | | 18 |
|
Signature: | | [6,6] |
|
Discriminant: | |
2259436291848000000000
=212⋅324⋅59
|
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
|
Root discriminant: | | 15.36 | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
|
Galois root discriminant: | | 2⋅34/351/2≈19.349808478363364
|
Ramified primes: | |
2, 3, 5
|
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
|
Discriminant root field: | | Q(5)
|
Aut(K/Q):
| | C6 |
|
This field is not Galois over Q. |
This is not a CM field. |
1, a, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, 51a12−52a11−51a10+52a9+51a8+51a7+52a6+51a5+52a4−51a2−51a+51, 51a13−52a8−51a7−51a5−51a4−51a3+52a2−51a+52, 51a14−52a9−51a8−51a6−51a5−51a4+52a3−51a2+52a, 51a15−52a10−51a9−51a7−51a6−51a5+52a4−51a3+52a2, 951a16−952a15+952a13+957a12+9534a11+9531a10−954a9+9542a8+9526a7−196a6−9511a5−9533a4−9528a3+9533a2−9514a+9546, 15194137551a17−15194137557394643a16−15194137559466821a15−1519413755149132176a14−151941375541528887a13+1519413755107213891a12+33839951524261a11−1519413755285350033a10−151941375578261727a9−30388275119149656a8+7996914525003882a7−151941375570798287a6−30388275126278124a5−84883452963992a4+303882751121277203a3+1519413755197376302a2+1519413755196061727a−1519413755688023346
Trivial group, which has order 1
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank: | | 11
|
|
Torsion generator: | |
−1
(order 2)
| sage: UK.torsion_generator()
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
|
Fundamental units: | |
84883454222683a17−848834523453349a16+169766910773565a15−848834531014309a14−8488345171242124a13+8488345572325528a12−189052012187a11+893517482993a10+8488345258117323a9−84883451621748478a8+84883452642417379a7−84883452893930678a6+84883452168067801a5−8488345858726669a4+84883459573743a3+848834593301107a2−84883453520803a+84883457103231, 15194137551155795243a17−79969145352478243a16+151941375515816010774a15−151941375510021131009a14−3038827519614772265a13+30388275133284234222a12−3383995593590219a11+1519413755214501841088a10+151941375566611768178a9−1519413755470044427701a8+1519413755773724994636a7−303882751171102314172a6+1519413755654997908834a5−84883451471449282a4+15194137559904640809a3+151941375520207304576a2−15194137553946737999a+15194137551955710076, 1519413755568069924a17−79969145119727119a16+15194137552820116587a15+15194137554574678414a14−151941375524297220148a13+151941375539772316039a12−338399550126699a11−3038827517018081201a10+30388275120672183144a9−1519413755117373388539a8+30388275113733111757a7−15194137554664085548a6−151941375581033920627a5+8488345594723819a4−151941375528622931163a3−151941375515048125443a2+15194137558499263159a+151941375531373481, 303882751236667436a17−1599382962710694a16+151941375511829105047a15−15194137551848172533a14−151941375549888763347a13+1519413755133369018021a12−3383995393877358a11+30388275118932985192a10+1519413755131257997676a9−1519413755380724783877a8+1519413755515090494031a7−1519413755505829451147a6+1519413755313905088444a5−8488345343170751a4−151941375517592162336a3−15194137556444751798a2+303882751973624470a+15194137552134059582, 30388275168723821a17−303882751375177022a16+15194137554337088624a15−15194137552767260223a14−151941375512875701727a13+151941375545123328849a12−3383995166555537a11+151941375565700277458a10+3038827511466582704a9−1519413755121024888454a8+1519413755219121228452a7−1519413755258595149833a6+159938292213105208a5−8488345584002851a4+151941375523057413352a3+15194137557364474443a2−15194137556946270218a−303882751105542892, 1519413755943859392a17−15194137555293788257a16+151941375512164386259a15−15194137556792436688a14−799691452070891002a13+799691456834011393a12−67679989893378a11+1519413755152929367903a10+151941375568832149792a9−30388275174422776540a8+303882751117572783523a7−1519413755630237538423a6+1519413755462512967337a5−8488345931783417a4−151941375512019445851a3+151941375517313850099a2+79969145127984454a+303882751150414986, 1519413755615267256a17−79969145193826351a16+3038827511801797692a15−15194137556422850947a14−151941375525591543199a13+151941375593596701982a12−3383995343750659a11+30388275125951066393a10+151941375527897081956a9−1519413755263351395338a8+1519413755446424935246a7−1519413755501165365599a6+1519413755394939009071a5−1697669187578914a4+151941375511030768827a3+3038827511720674729a2−15194137553631140809a+15194137552102686101, 1519413755385937328a17−15194137552414200761a16+15194137556192819684a15−15194137555167093322a14−151941375515719421344a13+151941375563106878233a12−3383995243612231a11+151941375598909082279a10+1599382972540259a9−1519413755175773691928a8+1519413755315580690612a7−1519413755364351482387a6+1519413755298423866326a5−8488345782585404a4+151941375517289011824a3+79969145286636282a2−1519413755794027949a+15194137552600566206, 1519413755250163492a17−15194137551301380191a16+15194137552633481191a15−1519413755701083448a14−3038827512045737935a13+151941375528732638013a12−338399589939446a11+151941375527362531207a10+151941375518189837392a9−151941375575063557856a8+30388275123075046728a7−1519413755130496717514a6+30388275119700081490a5−8488345244932949a4+151941375520276024326a3−15194137556648222162a2−15194137551232425118a−1519413755375682798, 1519413755677435809a17−15194137553824360526a16+15194137558710474833a15−15194137554731554201a14−799691451488402243a13+799691454862180987a12−3383995318139026a11+1519413755109766441069a10+151941375543241227382a9−1519413755254413685712a8+1519413755409250118312a7−1519413755453815688071a6+30388275170023481634a5−1697669165012905a4+3038827515671962800a3−151941375510270810403a2+79969145155301771a+15194137551130988692, 15194137551046015209a17−3038827511272290328a16+3038827513116531750a15−3038827512211792690a14−151941375544612160319a13+1519413755162375460322a12−3383995591748171a11+30388275143918220075a10+151941375556528251501a9−1519413755461872020776a8+303882751153609521877a7−1519413755851467345132a6+303882751131983921186a5−44675577929298a4+15194137553555954114a3+151941375520706362617a2−1519413755670940181a+15194137552769890892
| sage: UK.fundamental_units()
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
|
Regulator: | | 4306.402696834701
|
|
s→1lim(s−1)ζK(s)=(≈(≈(w⋅∣D∣2r1⋅(2π)r2⋅R⋅h2⋅225943629184800000000026⋅(2π)6⋅4306.402696834701⋅10.178379021437612
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
C6×S3 (as 18T6):
sage: K.galois_group(type='pari')
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
p |
2 |
3 |
5 |
7 |
11 |
13 |
17 |
19 |
23 |
29 |
31 |
37 |
41 |
43 |
47 |
53 |
59 |
Cycle type |
R |
R |
R |
63 |
62,32 |
63 |
29 |
26,16 |
63 |
36 |
62,32 |
29 |
36 |
63 |
63 |
29 |
62,32 |
In the table, R denotes a ramified prime.
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
# to obtain a list of
[ei,fi] for the factorization of the ideal
pOK for
p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of
[ei,fi] for the factorization of the ideal
pOK for
p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of
[ei,fi] for the factorization of the ideal
pOK for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of
[ei,fi] for the factorization of the ideal
pOK for
p=7 in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
p | Label | Polynomial
| e |
f |
c |
Galois group |
Slope content |
2
| 2.6.1.0a1.1 | x6+x4+x3+x+1 | 1 | 6 | 0 | C6 | [ ]6 |
2.6.2.12a1.1 | x12+2x10+2x9+x8+4x7+5x6+2x5+6x4+4x3+x2+4x+5 | 2 | 6 | 12 | C6×C2 | [2]6 |
3
| 3.6.3.24a2.1 | x18+6x16+15x14+6x13+32x12+24x11+63x10+36x9+90x8+72x7+109x6+102x5+96x4+56x3+84x2+72x+35 | 3 | 6 | 24 | not computed | not computed |
5
| 5.3.2.3a1.2 | x6+6x4+6x3+9x2+18x+14 | 2 | 3 | 3 | C6 | [ ]23 |
5.3.2.3a1.2 | x6+6x4+6x3+9x2+18x+14 | 2 | 3 | 3 | C6 | [ ]23 |
5.3.2.3a1.2 | x6+6x4+6x3+9x2+18x+14 | 2 | 3 | 3 | C6 | [ ]23 |