LMFDB - Number field 18.6.49048530062408000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 1)

gp: K = bnfinit(y^18 - 2*y^17 - 6*y^16 + 17*y^15 + 30*y^14 - 76*y^13 - 72*y^12 + 210*y^11 + 106*y^10 - 303*y^9 - 118*y^8 + 262*y^7 + 102*y^6 - 142*y^5 - 68*y^4 + 25*y^3 + 16*y^2 - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 1)

$ x^{18} - 2 x^{17} - 6 x^{16} + 17 x^{15} + 30 x^{14} - 76 x^{13} - 72 x^{12} + 210 x^{11} + 106 x^{10} + \cdots - 1 $ x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$18$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[6, 6]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$49048530062408000000000$ 49048530062408000000000 $\medspace = 2^{12}\cdot 5^{9}\cdot 19^{10}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$18.22$	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:	$2^{2/3}5^{1/2}19^{5/6}\approx 41.28570135662403$
Ramified primes:	$2$, $5$, $19$ 2, 5, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{5}) $
$\card{ \Aut(K/\Q) }$:	$3$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19}a^{15}-\frac{8}{19}a^{14}-\frac{8}{19}a^{13}+\frac{1}{19}a^{12}-\frac{6}{19}a^{11}-\frac{7}{19}a^{10}-\frac{4}{19}a^{9}+\frac{5}{19}a^{8}+\frac{9}{19}a^{7}-\frac{5}{19}a^{6}-\frac{8}{19}a^{5}-\frac{6}{19}a^{4}+\frac{8}{19}a^{3}+\frac{3}{19}a^{2}-\frac{1}{19}a+\frac{7}{19}$, $\frac{1}{19}a^{16}+\frac{4}{19}a^{14}-\frac{6}{19}a^{13}+\frac{2}{19}a^{12}+\frac{2}{19}a^{11}-\frac{3}{19}a^{10}-\frac{8}{19}a^{9}-\frac{8}{19}a^{8}-\frac{9}{19}a^{7}+\frac{9}{19}a^{6}+\frac{6}{19}a^{5}-\frac{2}{19}a^{4}-\frac{9}{19}a^{3}+\frac{4}{19}a^{2}-\frac{1}{19}a-\frac{1}{19}$, $\frac{1}{3575415686411}a^{17}-\frac{22069974359}{3575415686411}a^{16}-\frac{83854298401}{3575415686411}a^{15}-\frac{1468873617460}{3575415686411}a^{14}+\frac{999781043856}{3575415686411}a^{13}+\frac{966963651589}{3575415686411}a^{12}+\frac{635433649389}{3575415686411}a^{11}+\frac{761486975039}{3575415686411}a^{10}+\frac{976633283357}{3575415686411}a^{9}+\frac{825989932961}{3575415686411}a^{8}-\frac{954594375449}{3575415686411}a^{7}-\frac{414352237697}{3575415686411}a^{6}+\frac{973185113821}{3575415686411}a^{5}+\frac{603898036870}{3575415686411}a^{4}-\frac{75090234302}{3575415686411}a^{3}+\frac{55635046609}{3575415686411}a^{2}+\frac{974491859567}{3575415686411}a+\frac{194716705441}{3575415686411}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, 1/19*a^15 - 8/19*a^14 - 8/19*a^13 + 1/19*a^12 - 6/19*a^11 - 7/19*a^10 - 4/19*a^9 + 5/19*a^8 + 9/19*a^7 - 5/19*a^6 - 8/19*a^5 - 6/19*a^4 + 8/19*a^3 + 3/19*a^2 - 1/19*a + 7/19, 1/19*a^16 + 4/19*a^14 - 6/19*a^13 + 2/19*a^12 + 2/19*a^11 - 3/19*a^10 - 8/19*a^9 - 8/19*a^8 - 9/19*a^7 + 9/19*a^6 + 6/19*a^5 - 2/19*a^4 - 9/19*a^3 + 4/19*a^2 - 1/19*a - 1/19, 1/3575415686411*a^17 - 22069974359/3575415686411*a^16 - 83854298401/3575415686411*a^15 - 1468873617460/3575415686411*a^14 + 999781043856/3575415686411*a^13 + 966963651589/3575415686411*a^12 + 635433649389/3575415686411*a^11 + 761486975039/3575415686411*a^10 + 976633283357/3575415686411*a^9 + 825989932961/3575415686411*a^8 - 954594375449/3575415686411*a^7 - 414352237697/3575415686411*a^6 + 973185113821/3575415686411*a^5 + 603898036870/3575415686411*a^4 - 75090234302/3575415686411*a^3 + 55635046609/3575415686411*a^2 + 974491859567/3575415686411*a + 194716705441/3575415686411

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{206517628}{1863166069}a^{17}-\frac{69640076}{1863166069}a^{16}-\frac{2060704690}{1863166069}a^{15}+\frac{1951719068}{1863166069}a^{14}+\frac{12209269886}{1863166069}a^{13}-\frac{8623285083}{1863166069}a^{12}-\frac{40399384520}{1863166069}a^{11}+\frac{32553010316}{1863166069}a^{10}+\frac{83485365488}{1863166069}a^{9}-\frac{56893192006}{1863166069}a^{8}-\frac{92996672072}{1863166069}a^{7}+\frac{43823692268}{1863166069}a^{6}+\frac{66555088158}{1863166069}a^{5}-\frac{14439310414}{1863166069}a^{4}-\frac{32612468246}{1863166069}a^{3}-\frac{8439017410}{1863166069}a^{2}+\frac{1277448896}{1863166069}a+\frac{1824833220}{1863166069}$, $\frac{1466673475920}{3575415686411}a^{17}-\frac{4393252640394}{3575415686411}a^{16}-\frac{5808787440837}{3575415686411}a^{15}+\frac{33520648246239}{3575415686411}a^{14}+\frac{18729939204981}{3575415686411}a^{13}-\frac{153771555971619}{3575415686411}a^{12}+\frac{7689827559546}{3575415686411}a^{11}+\frac{405490901693709}{3575415686411}a^{10}-\frac{157633831809088}{3575415686411}a^{9}-\frac{575811725854344}{3575415686411}a^{8}+\frac{280991708643807}{3575415686411}a^{7}+\frac{511014010795278}{3575415686411}a^{6}-\frac{248175249401403}{3575415686411}a^{5}-\frac{299668866322935}{3575415686411}a^{4}+\frac{116016178153188}{3575415686411}a^{3}+\frac{92663007187722}{3575415686411}a^{2}-\frac{16536704195520}{3575415686411}a-\frac{7263339588861}{3575415686411}$, $a$, $\frac{60763858954}{3575415686411}a^{17}-\frac{921600391414}{3575415686411}a^{16}+\frac{1866554168515}{3575415686411}a^{15}+\frac{4264763602909}{3575415686411}a^{14}-\frac{14661140221852}{3575415686411}a^{13}-\frac{16730900450288}{3575415686411}a^{12}+\frac{68720093422364}{3575415686411}a^{11}+\frac{18509192546581}{3575415686411}a^{10}-\frac{177387155384006}{3575415686411}a^{9}+\frac{28388145055671}{3575415686411}a^{8}+\frac{227654078999846}{3575415686411}a^{7}-\frac{53882042899938}{3575415686411}a^{6}-\frac{182186152887450}{3575415686411}a^{5}+\frac{37559424662272}{3575415686411}a^{4}+\frac{96183831017719}{3575415686411}a^{3}-\frac{8070178533518}{3575415686411}a^{2}-\frac{18822162262065}{3575415686411}a-\frac{2010182712857}{3575415686411}$, $\frac{1460871916750}{3575415686411}a^{17}-\frac{3834688352322}{3575415686411}a^{16}-\frac{6945315807773}{3575415686411}a^{15}+\frac{30337119140111}{3575415686411}a^{14}+\frac{28335803435920}{3575415686411}a^{13}-\frac{138649726441477}{3575415686411}a^{12}-\frac{35763657922400}{3575415686411}a^{11}+\frac{373589253726833}{3575415686411}a^{10}-\frac{37451150952978}{3575415686411}a^{9}-\frac{542207901447878}{3575415686411}a^{8}+\frac{107719929862969}{3575415686411}a^{7}+\frac{492904382399191}{3575415686411}a^{6}-\frac{96731907774448}{3575415686411}a^{5}-\frac{296657754915037}{3575415686411}a^{4}+\frac{36736898542853}{3575415686411}a^{3}+\frac{90808328720313}{3575415686411}a^{2}-\frac{3223435479741}{3575415686411}a-\frac{7288359841552}{3575415686411}$, $\frac{115999209185}{3575415686411}a^{17}-\frac{1101237385580}{3575415686411}a^{16}+\frac{1588446151764}{3575415686411}a^{15}+\frac{5512531917559}{3575415686411}a^{14}-\frac{12890839416193}{3575415686411}a^{13}-\frac{23479912409992}{3575415686411}a^{12}+\frac{61988803125031}{3575415686411}a^{11}+\frac{38208488454154}{3575415686411}a^{10}-\frac{156795545333577}{3575415686411}a^{9}-\frac{15375585936803}{3575415686411}a^{8}+\frac{9445516785021}{188179772969}a^{7}+\frac{17885643240822}{3575415686411}a^{6}-\frac{131228010343876}{3575415686411}a^{5}-\frac{29007603284604}{3575415686411}a^{4}+\frac{66796204079158}{3575415686411}a^{3}+\frac{21637364355912}{3575415686411}a^{2}-\frac{5639612048366}{3575415686411}a-\frac{4144575308926}{3575415686411}$, $\frac{451542678363}{3575415686411}a^{17}-\frac{313276300010}{3575415686411}a^{16}-\frac{4232600316861}{3575415686411}a^{15}+\frac{4993117206142}{3575415686411}a^{14}+\frac{25199889716893}{3575415686411}a^{13}-\frac{23297096033981}{3575415686411}a^{12}-\frac{84257709191213}{3575415686411}a^{11}+\frac{4324659089683}{188179772969}a^{10}+\frac{180800026421901}{3575415686411}a^{9}-\frac{152941766451988}{3575415686411}a^{8}-\frac{226649873790615}{3575415686411}a^{7}+\frac{155865351603052}{3575415686411}a^{6}+\frac{178677356718776}{3575415686411}a^{5}-\frac{94276064631342}{3575415686411}a^{4}-\frac{91970953502635}{3575415686411}a^{3}+\frac{13513068479640}{3575415686411}a^{2}+\frac{12058558958712}{3575415686411}a+\frac{1367462353111}{3575415686411}$, $\frac{1326943206623}{3575415686411}a^{17}-\frac{3885382248252}{3575415686411}a^{16}-\frac{4923639852998}{3575415686411}a^{15}+\frac{28238966502014}{3575415686411}a^{14}+\frac{16909240370178}{3575415686411}a^{13}-\frac{125766707287901}{3575415686411}a^{12}+\frac{4596524979513}{3575415686411}a^{11}+\frac{314511924516275}{3575415686411}a^{10}-\frac{112847528061098}{3575415686411}a^{9}-\frac{405080088099338}{3575415686411}a^{8}+\frac{164466074630131}{3575415686411}a^{7}+\frac{339675103854174}{3575415686411}a^{6}-\frac{120769872665770}{3575415686411}a^{5}-\frac{193187010687385}{3575415686411}a^{4}+\frac{43569502621270}{3575415686411}a^{3}+\frac{56292605561033}{3575415686411}a^{2}-\frac{3407886698044}{3575415686411}a-\frac{5261948336191}{3575415686411}$, $\frac{512096456742}{3575415686411}a^{17}-\frac{1877531929476}{3575415686411}a^{16}-\frac{909587320821}{3575415686411}a^{15}+\frac{12452754378508}{3575415686411}a^{14}-\frac{695309155199}{3575415686411}a^{13}-\frac{54626419555959}{3575415686411}a^{12}+\frac{32740318390516}{3575415686411}a^{11}+\frac{124970040297224}{3575415686411}a^{10}-\frac{117907064021004}{3575415686411}a^{9}-\frac{135263342596888}{3575415686411}a^{8}+\frac{145367191667960}{3575415686411}a^{7}+\frac{92587625371559}{3575415686411}a^{6}-\frac{91978180209042}{3575415686411}a^{5}-\frac{1861937706264}{188179772969}a^{4}+\frac{28745191893445}{3575415686411}a^{3}+\frac{1409392643125}{3575415686411}a^{2}+\frac{1515558298727}{3575415686411}a+\frac{870336131410}{3575415686411}$, $\frac{1101506625224}{3575415686411}a^{17}-\frac{1847694492968}{3575415686411}a^{16}-\frac{7394063849439}{3575415686411}a^{15}+\frac{16824658750637}{3575415686411}a^{14}+\frac{39365140979228}{3575415686411}a^{13}-\frac{74916262561022}{3575415686411}a^{12}-\frac{106934108961367}{3575415686411}a^{11}+\frac{214391375007778}{3575415686411}a^{10}+\frac{188941422768318}{3575415686411}a^{9}-\frac{320597216209927}{3575415686411}a^{8}-\frac{222111218188392}{3575415686411}a^{7}+\frac{284547366977849}{3575415686411}a^{6}+\frac{174749800014866}{3575415686411}a^{5}-\frac{160998400174758}{3575415686411}a^{4}-\frac{96070634023383}{3575415686411}a^{3}+\frac{35310185508165}{3575415686411}a^{2}+\frac{12654021790330}{3575415686411}a-\frac{5969711640726}{3575415686411}$, $\frac{3607574690857}{3575415686411}a^{17}-\frac{6626446585844}{3575415686411}a^{16}-\frac{22012375935202}{3575415686411}a^{15}+\frac{2954283260717}{188179772969}a^{14}+\frac{113693121537423}{3575415686411}a^{13}-\frac{242969035204417}{3575415686411}a^{12}-\frac{282200894137078}{3575415686411}a^{11}+\frac{656356096174745}{3575415686411}a^{10}+\frac{457038501352420}{3575415686411}a^{9}-\frac{875125114072630}{3575415686411}a^{8}-\frac{539571529012283}{3575415686411}a^{7}+\frac{672737668989262}{3575415686411}a^{6}+\frac{451651752690754}{3575415686411}a^{5}-\frac{296212021468830}{3575415686411}a^{4}-\frac{266170202801095}{3575415686411}a^{3}-\frac{23857473451502}{3575415686411}a^{2}+\frac{32205965376077}{3575415686411}a+\frac{10171143124002}{3575415686411}$ 206517628/1863166069a^17 - 69640076/1863166069a^16 - 2060704690/1863166069a^15 + 1951719068/1863166069a^14 + 12209269886/1863166069a^13 - 8623285083/1863166069a^12 - 40399384520/1863166069a^11 + 32553010316/1863166069a^10 + 83485365488/1863166069a^9 - 56893192006/1863166069a^8 - 92996672072/1863166069a^7 + 43823692268/1863166069a^6 + 66555088158/1863166069a^5 - 14439310414/1863166069a^4 - 32612468246/1863166069a^3 - 8439017410/1863166069a^2 + 1277448896/1863166069a + 1824833220/1863166069, 1466673475920/3575415686411a^17 - 4393252640394/3575415686411a^16 - 5808787440837/3575415686411a^15 + 33520648246239/3575415686411a^14 + 18729939204981/3575415686411a^13 - 153771555971619/3575415686411a^12 + 7689827559546/3575415686411a^11 + 405490901693709/3575415686411a^10 - 157633831809088/3575415686411a^9 - 575811725854344/3575415686411a^8 + 280991708643807/3575415686411a^7 + 511014010795278/3575415686411a^6 - 248175249401403/3575415686411a^5 - 299668866322935/3575415686411a^4 + 116016178153188/3575415686411a^3 + 92663007187722/3575415686411a^2 - 16536704195520/3575415686411a - 7263339588861/3575415686411, a, 60763858954/3575415686411a^17 - 921600391414/3575415686411a^16 + 1866554168515/3575415686411a^15 + 4264763602909/3575415686411a^14 - 14661140221852/3575415686411a^13 - 16730900450288/3575415686411a^12 + 68720093422364/3575415686411a^11 + 18509192546581/3575415686411a^10 - 177387155384006/3575415686411a^9 + 28388145055671/3575415686411a^8 + 227654078999846/3575415686411a^7 - 53882042899938/3575415686411a^6 - 182186152887450/3575415686411a^5 + 37559424662272/3575415686411a^4 + 96183831017719/3575415686411a^3 - 8070178533518/3575415686411a^2 - 18822162262065/3575415686411a - 2010182712857/3575415686411, 1460871916750/3575415686411a^17 - 3834688352322/3575415686411a^16 - 6945315807773/3575415686411a^15 + 30337119140111/3575415686411a^14 + 28335803435920/3575415686411a^13 - 138649726441477/3575415686411a^12 - 35763657922400/3575415686411a^11 + 373589253726833/3575415686411a^10 - 37451150952978/3575415686411a^9 - 542207901447878/3575415686411a^8 + 107719929862969/3575415686411a^7 + 492904382399191/3575415686411a^6 - 96731907774448/3575415686411a^5 - 296657754915037/3575415686411a^4 + 36736898542853/3575415686411a^3 + 90808328720313/3575415686411a^2 - 3223435479741/3575415686411a - 7288359841552/3575415686411, 115999209185/3575415686411a^17 - 1101237385580/3575415686411a^16 + 1588446151764/3575415686411a^15 + 5512531917559/3575415686411a^14 - 12890839416193/3575415686411a^13 - 23479912409992/3575415686411a^12 + 61988803125031/3575415686411a^11 + 38208488454154/3575415686411a^10 - 156795545333577/3575415686411a^9 - 15375585936803/3575415686411a^8 + 9445516785021/188179772969a^7 + 17885643240822/3575415686411a^6 - 131228010343876/3575415686411a^5 - 29007603284604/3575415686411a^4 + 66796204079158/3575415686411a^3 + 21637364355912/3575415686411a^2 - 5639612048366/3575415686411a - 4144575308926/3575415686411, 451542678363/3575415686411a^17 - 313276300010/3575415686411a^16 - 4232600316861/3575415686411a^15 + 4993117206142/3575415686411a^14 + 25199889716893/3575415686411a^13 - 23297096033981/3575415686411a^12 - 84257709191213/3575415686411a^11 + 4324659089683/188179772969a^10 + 180800026421901/3575415686411a^9 - 152941766451988/3575415686411a^8 - 226649873790615/3575415686411a^7 + 155865351603052/3575415686411a^6 + 178677356718776/3575415686411a^5 - 94276064631342/3575415686411a^4 - 91970953502635/3575415686411a^3 + 13513068479640/3575415686411a^2 + 12058558958712/3575415686411a + 1367462353111/3575415686411, 1326943206623/3575415686411a^17 - 3885382248252/3575415686411a^16 - 4923639852998/3575415686411a^15 + 28238966502014/3575415686411a^14 + 16909240370178/3575415686411a^13 - 125766707287901/3575415686411a^12 + 4596524979513/3575415686411a^11 + 314511924516275/3575415686411a^10 - 112847528061098/3575415686411a^9 - 405080088099338/3575415686411a^8 + 164466074630131/3575415686411a^7 + 339675103854174/3575415686411a^6 - 120769872665770/3575415686411a^5 - 193187010687385/3575415686411a^4 + 43569502621270/3575415686411a^3 + 56292605561033/3575415686411a^2 - 3407886698044/3575415686411a - 5261948336191/3575415686411, 512096456742/3575415686411a^17 - 1877531929476/3575415686411a^16 - 909587320821/3575415686411a^15 + 12452754378508/3575415686411a^14 - 695309155199/3575415686411a^13 - 54626419555959/3575415686411a^12 + 32740318390516/3575415686411a^11 + 124970040297224/3575415686411a^10 - 117907064021004/3575415686411a^9 - 135263342596888/3575415686411a^8 + 145367191667960/3575415686411a^7 + 92587625371559/3575415686411a^6 - 91978180209042/3575415686411a^5 - 1861937706264/188179772969a^4 + 28745191893445/3575415686411a^3 + 1409392643125/3575415686411a^2 + 1515558298727/3575415686411a + 870336131410/3575415686411, 1101506625224/3575415686411a^17 - 1847694492968/3575415686411a^16 - 7394063849439/3575415686411a^15 + 16824658750637/3575415686411a^14 + 39365140979228/3575415686411a^13 - 74916262561022/3575415686411a^12 - 106934108961367/3575415686411a^11 + 214391375007778/3575415686411a^10 + 188941422768318/3575415686411a^9 - 320597216209927/3575415686411a^8 - 222111218188392/3575415686411a^7 + 284547366977849/3575415686411a^6 + 174749800014866/3575415686411a^5 - 160998400174758/3575415686411a^4 - 96070634023383/3575415686411a^3 + 35310185508165/3575415686411a^2 + 12654021790330/3575415686411a - 5969711640726/3575415686411, 3607574690857/3575415686411a^17 - 6626446585844/3575415686411a^16 - 22012375935202/3575415686411a^15 + 2954283260717/188179772969a^14 + 113693121537423/3575415686411a^13 - 242969035204417/3575415686411a^12 - 282200894137078/3575415686411a^11 + 656356096174745/3575415686411a^10 + 457038501352420/3575415686411a^9 - 875125114072630/3575415686411a^8 - 539571529012283/3575415686411a^7 + 672737668989262/3575415686411a^6 + 451651752690754/3575415686411a^5 - 296212021468830/3575415686411a^4 - 266170202801095/3575415686411a^3 - 23857473451502/3575415686411a^2 + 32205965376077/3575415686411a + 10171143124002/3575415686411	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 20286.0811613 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 20286.0811613 \cdot 1}{2\cdot\sqrt{49048530062408000000000}}\cr\approx \mathstrut & 0.180349253447 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 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 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 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(QQ); K<a> := NumberField(x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 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 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 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))))

Galois group

$C_3\times S_3^2$ (as 18T46):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 108

The 27 conjugacy class representatives for $C_3\times S_3^2$

Character table for $C_3\times S_3^2$

Intermediate fields

$\Q(\sqrt{5}) $, 3.1.76.1, 6.2.722000.1, 6.6.722000.1

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]

Sibling fields

Degree 12 sibling:	data not computed
Degree 18 siblings:	data not computed
Degree 27 sibling:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	12.0.4245890760250000.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.6.0.1}{6} }^{3}$	R	${\href{/padicField/7.6.0.1}{6} }^{3}$	${\href{/padicField/11.3.0.1}{3} }^{6}$	${\href{/padicField/13.6.0.1}{6} }^{3}$	${\href{/padicField/17.6.0.1}{6} }^{3}$	R	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$	${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$	${\href{/padicField/43.6.0.1}{6} }^{3}$	${\href{/padicField/47.6.0.1}{6} }^{3}$	${\href{/padicField/53.6.0.1}{6} }^{3}$	${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.18.12.1	$x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$	$3$	$6$	$12$	$S_3 \times C_3$	$[\ ]_{3}^{6}$
$5$ 5	5.6.3.1	$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.1	$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.1	$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$19$ 19	19.3.0.1	$x^{3} + 4 x + 17$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	19.3.2.1	$x^{3} + 76$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	19.6.3.2	$x^{6} + 65 x^{4} + 34 x^{3} + 1099 x^{2} - 1802 x + 4564$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	19.6.5.4	$x^{6} + 76$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$

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