LMFDB - Number field 18.6.19446011944726528000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*x + 1)

gp: K = bnfinit(y^18 - 6*y^17 + 11*y^16 + 6*y^15 - 60*y^14 + 122*y^13 - 141*y^12 + 74*y^11 + 54*y^10 - 122*y^9 + 71*y^8 + 36*y^7 - 70*y^6 + 16*y^5 + 33*y^4 - 10*y^3 - 11*y^2 - 4*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*x + 1);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*x + 1)

$x^{18} - 6 x^{17} + 11 x^{16} + 6 x^{15} - 60 x^{14} + 122 x^{13} - 141 x^{12} + 74 x^{11} + 54 x^{10} + \cdots + 1$ x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*x + 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:	$19446011944726528000000$ 19446011944726528000000 $\medspace = 2^{18}\cdot 5^{6}\cdot 7^{15}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$17.31$	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:	$2\cdot 5^{1/2}7^{5/6}\approx 22.634106993721137$
Ramified primes:	$2$ , $5$ , $7$ 2, 5, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{7})$
$\Aut(K/\Q)$ :	$C_6$	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 $a$ )

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $a^{6}$ , $a^{7}$ , $a^{8}$ , $a^{9}$ , $a^{10}$ , $a^{11}$ , $a^{12}$ , $a^{13}$ , $a^{14}$ , $\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}$ , $\frac{1}{3}a^{16}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$ , $\frac{1}{86029642954203}a^{17}-\frac{869649721279}{6617664842631}a^{16}+\frac{7235668926211}{86029642954203}a^{15}-\frac{7506573131436}{28676547651401}a^{14}+\frac{34538291624156}{86029642954203}a^{13}-\frac{23411632094356}{86029642954203}a^{12}+\frac{3114905726745}{28676547651401}a^{11}-\frac{11118832043938}{86029642954203}a^{10}-\frac{14367304156966}{86029642954203}a^{9}+\frac{38301130535374}{86029642954203}a^{8}+\frac{14666286105091}{86029642954203}a^{7}+\frac{8375323339453}{86029642954203}a^{6}+\frac{5123296022740}{86029642954203}a^{5}-\frac{6937657893167}{28676547651401}a^{4}+\frac{5205403477032}{28676547651401}a^{3}-\frac{38217903043877}{86029642954203}a^{2}-\frac{14308576464381}{28676547651401}a+\frac{12557213140464}{28676547651401}$ 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, 1/3*a^15 - 1/3*a^14 + 1/3*a^13 - 1/3*a^11 - 1/3*a^10 - 1/3*a^8 - 1/3*a^7 + 1/3*a^6 + 1/3*a^5 + 1/3*a^4 + 1/3*a^3 + 1/3, 1/3*a^16 + 1/3*a^13 - 1/3*a^12 + 1/3*a^11 - 1/3*a^10 - 1/3*a^9 + 1/3*a^8 - 1/3*a^6 - 1/3*a^5 - 1/3*a^4 + 1/3*a^3 + 1/3*a + 1/3, 1/86029642954203*a^17 - 869649721279/6617664842631*a^16 + 7235668926211/86029642954203*a^15 - 7506573131436/28676547651401*a^14 + 34538291624156/86029642954203*a^13 - 23411632094356/86029642954203*a^12 + 3114905726745/28676547651401*a^11 - 11118832043938/86029642954203*a^10 - 14367304156966/86029642954203*a^9 + 38301130535374/86029642954203*a^8 + 14666286105091/86029642954203*a^7 + 8375323339453/86029642954203*a^6 + 5123296022740/86029642954203*a^5 - 6937657893167/28676547651401*a^4 + 5205403477032/28676547651401*a^3 - 38217903043877/86029642954203*a^2 - 14308576464381/28676547651401*a + 12557213140464/28676547651401

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:	$\frac{3289542082228}{86029642954203}a^{17}-\frac{1799757701782}{6617664842631}a^{16}+\frac{55028069236597}{86029642954203}a^{15}-\frac{1500705654290}{28676547651401}a^{14}-\frac{237659493297832}{86029642954203}a^{13}+\frac{574520026251833}{86029642954203}a^{12}-\frac{249489167590787}{28676547651401}a^{11}+\frac{555080384696768}{86029642954203}a^{10}+\frac{32879807823350}{86029642954203}a^{9}-\frac{538733483242271}{86029642954203}a^{8}+\frac{470862843352528}{86029642954203}a^{7}-\frac{103227648218336}{86029642954203}a^{6}-\frac{72544003371644}{86029642954203}a^{5}-\frac{18642421980662}{28676547651401}a^{4}+\frac{57040437212715}{28676547651401}a^{3}+\frac{594280176319}{86029642954203}a^{2}-\frac{37901077484942}{28676547651401}a-\frac{13882749489042}{28676547651401}$ , $\frac{10107444161534}{86029642954203}a^{17}-\frac{1310689001135}{2205888280877}a^{16}+\frac{17537457616095}{28676547651401}a^{15}+\frac{175103872114817}{86029642954203}a^{14}-\frac{569298995253299}{86029642954203}a^{13}+\frac{652011715389416}{86029642954203}a^{12}-\frac{151444481286266}{86029642954203}a^{11}-\frac{814379850117029}{86029642954203}a^{10}+\frac{14\!\cdots\!43}{86029642954203}a^{9}-\frac{253631069656090}{28676547651401}a^{8}-\frac{711567328958150}{86029642954203}a^{7}+\frac{14\!\cdots\!60}{86029642954203}a^{6}-\frac{535929923429204}{86029642954203}a^{5}-\frac{608595558301057}{86029642954203}a^{4}+\frac{240621300107278}{28676547651401}a^{3}+\frac{200603290735637}{86029642954203}a^{2}-\frac{332668600636666}{86029642954203}a-\frac{30017238356392}{28676547651401}$ , $\frac{74650376}{9891875699}a^{17}-\frac{1777332580}{29675627097}a^{16}+\frac{7363630100}{29675627097}a^{15}-\frac{17336760668}{29675627097}a^{14}+\frac{10670882644}{29675627097}a^{13}+\frac{62280325111}{29675627097}a^{12}-\frac{72650552096}{9891875699}a^{11}+\frac{370306137974}{29675627097}a^{10}-\frac{391261400792}{29675627097}a^{9}+\frac{66568649411}{9891875699}a^{8}+\frac{112092967180}{29675627097}a^{7}-\frac{90341064076}{9891875699}a^{6}+\frac{50264912844}{9891875699}a^{5}+\frac{26212688363}{9891875699}a^{4}-\frac{158226548072}{29675627097}a^{3}+\frac{18799168058}{9891875699}a^{2}+\frac{61433386916}{29675627097}a-\frac{27683385680}{29675627097}$ , $\frac{2088287592}{9891875699}a^{17}-\frac{34977325742}{29675627097}a^{16}+\frac{52470378560}{29675627097}a^{15}+\frac{71272482202}{29675627097}a^{14}-\frac{123602695158}{9891875699}a^{13}+\frac{606323429582}{29675627097}a^{12}-\frac{511906512268}{29675627097}a^{11}-\frac{9612171614}{9891875699}a^{10}+\frac{679569529268}{29675627097}a^{9}-\frac{686163569674}{29675627097}a^{8}+\frac{22721291710}{29675627097}a^{7}+\frac{634603152067}{29675627097}a^{6}-\frac{522999117230}{29675627097}a^{5}-\frac{84637216067}{29675627097}a^{4}+\frac{123505764802}{9891875699}a^{3}-\frac{16766685651}{9891875699}a^{2}-\frac{112958611532}{29675627097}a-\frac{6092800099}{9891875699}$ , $\frac{8651079986890}{28676547651401}a^{17}-\frac{11694881464817}{6617664842631}a^{16}+\frac{89145352614793}{28676547651401}a^{15}+\frac{58126855775789}{28676547651401}a^{14}-\frac{15\!\cdots\!19}{86029642954203}a^{13}+\frac{30\!\cdots\!27}{86029642954203}a^{12}-\frac{34\!\cdots\!26}{86029642954203}a^{11}+\frac{17\!\cdots\!17}{86029642954203}a^{10}+\frac{14\!\cdots\!95}{86029642954203}a^{9}-\frac{32\!\cdots\!76}{86029642954203}a^{8}+\frac{689648349472216}{28676547651401}a^{7}+\frac{629147491123001}{86029642954203}a^{6}-\frac{16\!\cdots\!52}{86029642954203}a^{5}+\frac{636353941090256}{86029642954203}a^{4}+\frac{513662832723214}{86029642954203}a^{3}-\frac{64420431327545}{28676547651401}a^{2}-\frac{91941369888983}{86029642954203}a-\frac{180310182577403}{86029642954203}$ , $\frac{906541454023}{28676547651401}a^{17}-\frac{1333271784442}{6617664842631}a^{16}+\frac{14175969202583}{28676547651401}a^{15}-\frac{9641921250957}{28676547651401}a^{14}-\frac{124313901835078}{86029642954203}a^{13}+\frac{469363496072683}{86029642954203}a^{12}-\frac{845823352099135}{86029642954203}a^{11}+\frac{881439220791175}{86029642954203}a^{10}-\frac{415755771133367}{86029642954203}a^{9}-\frac{335553376947262}{86029642954203}a^{8}+\frac{272503805942910}{28676547651401}a^{7}-\frac{546998419713986}{86029642954203}a^{6}-\frac{14051577243578}{86029642954203}a^{5}+\frac{312001250339572}{86029642954203}a^{4}-\frac{84521021113612}{86029642954203}a^{3}-\frac{41057870319303}{28676547651401}a^{2}+\frac{179017447191197}{86029642954203}a+\frac{37077460646351}{86029642954203}$ , $\frac{916294797790}{28676547651401}a^{17}-\frac{241456815144}{2205888280877}a^{16}-\frac{5181770918704}{28676547651401}a^{15}+\frac{37991249766890}{28676547651401}a^{14}-\frac{52859138775899}{28676547651401}a^{13}-\frac{34164795129847}{28676547651401}a^{12}+\frac{216085338208427}{28676547651401}a^{11}-\frac{390778660702455}{28676547651401}a^{10}+\frac{397913170239529}{28676547651401}a^{9}-\frac{138683358538971}{28676547651401}a^{8}-\frac{158914298752714}{28676547651401}a^{7}+\frac{213832238008000}{28676547651401}a^{6}-\frac{9127710517037}{28676547651401}a^{5}-\frac{154209453726599}{28676547651401}a^{4}+\frac{69477622052192}{28676547651401}a^{3}+\frac{80225949934874}{28676547651401}a^{2}-\frac{34950514250917}{28676547651401}a-\frac{15849691165890}{28676547651401}$ , $\frac{2462409851133}{28676547651401}a^{17}-\frac{765876205647}{2205888280877}a^{16}+\frac{3158278497347}{86029642954203}a^{15}+\frac{162829633093033}{86029642954203}a^{14}-\frac{322155527504248}{86029642954203}a^{13}+\frac{56916211352037}{28676547651401}a^{12}+\frac{330197699807980}{86029642954203}a^{11}-\frac{10\!\cdots\!08}{86029642954203}a^{10}+\frac{408323433552594}{28676547651401}a^{9}-\frac{400506332570348}{86029642954203}a^{8}-\frac{549387035025980}{86029642954203}a^{7}+\frac{931741342410524}{86029642954203}a^{6}-\frac{175124960863444}{86029642954203}a^{5}-\frac{468265886760979}{86029642954203}a^{4}+\frac{420668605741151}{86029642954203}a^{3}+\frac{62956954412114}{28676547651401}a^{2}-\frac{37380940708646}{28676547651401}a-\frac{49379204214196}{86029642954203}$ , $\frac{2951818420362}{28676547651401}a^{17}-\frac{1579436660097}{2205888280877}a^{16}+\frac{154769277449699}{86029642954203}a^{15}-\frac{84363365764526}{86029642954203}a^{14}-\frac{481264572180898}{86029642954203}a^{13}+\frac{526670856791865}{28676547651401}a^{12}-\frac{27\!\cdots\!67}{86029642954203}a^{11}+\frac{29\!\cdots\!52}{86029642954203}a^{10}-\frac{540710591612818}{28676547651401}a^{9}-\frac{475953169958852}{86029642954203}a^{8}+\frac{18\!\cdots\!56}{86029642954203}a^{7}-\frac{15\!\cdots\!32}{86029642954203}a^{6}+\frac{255585881318399}{86029642954203}a^{5}+\frac{589601557249508}{86029642954203}a^{4}-\frac{491055860022328}{86029642954203}a^{3}+\frac{18790649378929}{28676547651401}a^{2}+\frac{28169930401567}{28676547651401}a-\frac{117963607151146}{86029642954203}$ , $\frac{10204689914066}{86029642954203}a^{17}-\frac{5035829106185}{6617664842631}a^{16}+\frac{136453187319245}{86029642954203}a^{15}+\frac{6455215340188}{28676547651401}a^{14}-\frac{639471520822124}{86029642954203}a^{13}+\frac{14\!\cdots\!57}{86029642954203}a^{12}-\frac{637190176066562}{28676547651401}a^{11}+\frac{12\!\cdots\!57}{86029642954203}a^{10}+\frac{296466185733433}{86029642954203}a^{9}-\frac{13\!\cdots\!69}{86029642954203}a^{8}+\frac{961904634169472}{86029642954203}a^{7}+\frac{464202423936059}{86029642954203}a^{6}-\frac{12\!\cdots\!60}{86029642954203}a^{5}+\frac{196943704554768}{28676547651401}a^{4}+\frac{132606232663362}{28676547651401}a^{3}-\frac{458458619477317}{86029642954203}a^{2}+\frac{25798828980231}{28676547651401}a+\frac{25038610288890}{28676547651401}$ , $\frac{1501062567904}{86029642954203}a^{17}-\frac{771607303784}{6617664842631}a^{16}+\frac{15606769062190}{86029642954203}a^{15}+\frac{12703287136105}{28676547651401}a^{14}-\frac{164732765071253}{86029642954203}a^{13}+\frac{65185296219122}{28676547651401}a^{12}+\frac{52843821157892}{86029642954203}a^{11}-\frac{174088246478180}{28676547651401}a^{10}+\frac{310258984982784}{28676547651401}a^{9}-\frac{245384771343460}{28676547651401}a^{8}-\frac{140489231310788}{86029642954203}a^{7}+\frac{836260291526558}{86029642954203}a^{6}-\frac{739466483441962}{86029642954203}a^{5}-\frac{372905951531}{86029642954203}a^{4}+\frac{504251724689654}{86029642954203}a^{3}-\frac{275531828235359}{86029642954203}a^{2}-\frac{191930130488392}{86029642954203}a+\frac{118307333344817}{86029642954203}$ 3289542082228/86029642954203a^17 - 1799757701782/6617664842631a^16 + 55028069236597/86029642954203a^15 - 1500705654290/28676547651401a^14 - 237659493297832/86029642954203a^13 + 574520026251833/86029642954203a^12 - 249489167590787/28676547651401a^11 + 555080384696768/86029642954203a^10 + 32879807823350/86029642954203a^9 - 538733483242271/86029642954203a^8 + 470862843352528/86029642954203a^7 - 103227648218336/86029642954203a^6 - 72544003371644/86029642954203a^5 - 18642421980662/28676547651401a^4 + 57040437212715/28676547651401a^3 + 594280176319/86029642954203a^2 - 37901077484942/28676547651401a - 13882749489042/28676547651401, 10107444161534/86029642954203a^17 - 1310689001135/2205888280877a^16 + 17537457616095/28676547651401a^15 + 175103872114817/86029642954203a^14 - 569298995253299/86029642954203a^13 + 652011715389416/86029642954203a^12 - 151444481286266/86029642954203a^11 - 814379850117029/86029642954203a^10 + 1470116033666843/86029642954203a^9 - 253631069656090/28676547651401a^8 - 711567328958150/86029642954203a^7 + 1426076171163760/86029642954203a^6 - 535929923429204/86029642954203a^5 - 608595558301057/86029642954203a^4 + 240621300107278/28676547651401a^3 + 200603290735637/86029642954203a^2 - 332668600636666/86029642954203a - 30017238356392/28676547651401, 74650376/9891875699a^17 - 1777332580/29675627097a^16 + 7363630100/29675627097a^15 - 17336760668/29675627097a^14 + 10670882644/29675627097a^13 + 62280325111/29675627097a^12 - 72650552096/9891875699a^11 + 370306137974/29675627097a^10 - 391261400792/29675627097a^9 + 66568649411/9891875699a^8 + 112092967180/29675627097a^7 - 90341064076/9891875699a^6 + 50264912844/9891875699a^5 + 26212688363/9891875699a^4 - 158226548072/29675627097a^3 + 18799168058/9891875699a^2 + 61433386916/29675627097a - 27683385680/29675627097, 2088287592/9891875699a^17 - 34977325742/29675627097a^16 + 52470378560/29675627097a^15 + 71272482202/29675627097a^14 - 123602695158/9891875699a^13 + 606323429582/29675627097a^12 - 511906512268/29675627097a^11 - 9612171614/9891875699a^10 + 679569529268/29675627097a^9 - 686163569674/29675627097a^8 + 22721291710/29675627097a^7 + 634603152067/29675627097a^6 - 522999117230/29675627097a^5 - 84637216067/29675627097a^4 + 123505764802/9891875699a^3 - 16766685651/9891875699a^2 - 112958611532/29675627097a - 6092800099/9891875699, 8651079986890/28676547651401a^17 - 11694881464817/6617664842631a^16 + 89145352614793/28676547651401a^15 + 58126855775789/28676547651401a^14 - 1506948086835119/86029642954203a^13 + 3005360870890127/86029642954203a^12 - 3454033968221726/86029642954203a^11 + 1745885815163717/86029642954203a^10 + 1477419490610195/86029642954203a^9 - 3208520743104476/86029642954203a^8 + 689648349472216/28676547651401a^7 + 629147491123001/86029642954203a^6 - 1659990019224052/86029642954203a^5 + 636353941090256/86029642954203a^4 + 513662832723214/86029642954203a^3 - 64420431327545/28676547651401a^2 - 91941369888983/86029642954203a - 180310182577403/86029642954203, 906541454023/28676547651401a^17 - 1333271784442/6617664842631a^16 + 14175969202583/28676547651401a^15 - 9641921250957/28676547651401a^14 - 124313901835078/86029642954203a^13 + 469363496072683/86029642954203a^12 - 845823352099135/86029642954203a^11 + 881439220791175/86029642954203a^10 - 415755771133367/86029642954203a^9 - 335553376947262/86029642954203a^8 + 272503805942910/28676547651401a^7 - 546998419713986/86029642954203a^6 - 14051577243578/86029642954203a^5 + 312001250339572/86029642954203a^4 - 84521021113612/86029642954203a^3 - 41057870319303/28676547651401a^2 + 179017447191197/86029642954203a + 37077460646351/86029642954203, 916294797790/28676547651401a^17 - 241456815144/2205888280877a^16 - 5181770918704/28676547651401a^15 + 37991249766890/28676547651401a^14 - 52859138775899/28676547651401a^13 - 34164795129847/28676547651401a^12 + 216085338208427/28676547651401a^11 - 390778660702455/28676547651401a^10 + 397913170239529/28676547651401a^9 - 138683358538971/28676547651401a^8 - 158914298752714/28676547651401a^7 + 213832238008000/28676547651401a^6 - 9127710517037/28676547651401a^5 - 154209453726599/28676547651401a^4 + 69477622052192/28676547651401a^3 + 80225949934874/28676547651401a^2 - 34950514250917/28676547651401a - 15849691165890/28676547651401, 2462409851133/28676547651401a^17 - 765876205647/2205888280877a^16 + 3158278497347/86029642954203a^15 + 162829633093033/86029642954203a^14 - 322155527504248/86029642954203a^13 + 56916211352037/28676547651401a^12 + 330197699807980/86029642954203a^11 - 1039273768574708/86029642954203a^10 + 408323433552594/28676547651401a^9 - 400506332570348/86029642954203a^8 - 549387035025980/86029642954203a^7 + 931741342410524/86029642954203a^6 - 175124960863444/86029642954203a^5 - 468265886760979/86029642954203a^4 + 420668605741151/86029642954203a^3 + 62956954412114/28676547651401a^2 - 37380940708646/28676547651401a - 49379204214196/86029642954203, 2951818420362/28676547651401a^17 - 1579436660097/2205888280877a^16 + 154769277449699/86029642954203a^15 - 84363365764526/86029642954203a^14 - 481264572180898/86029642954203a^13 + 526670856791865/28676547651401a^12 - 2717437779264467/86029642954203a^11 + 2942678797480252/86029642954203a^10 - 540710591612818/28676547651401a^9 - 475953169958852/86029642954203a^8 + 1826540188177456/86029642954203a^7 - 1529248174280932/86029642954203a^6 + 255585881318399/86029642954203a^5 + 589601557249508/86029642954203a^4 - 491055860022328/86029642954203a^3 + 18790649378929/28676547651401a^2 + 28169930401567/28676547651401a - 117963607151146/86029642954203, 10204689914066/86029642954203a^17 - 5035829106185/6617664842631a^16 + 136453187319245/86029642954203a^15 + 6455215340188/28676547651401a^14 - 639471520822124/86029642954203a^13 + 1481826458122957/86029642954203a^12 - 637190176066562/28676547651401a^11 + 1282035548142157/86029642954203a^10 + 296466185733433/86029642954203a^9 - 1385501749619569/86029642954203a^8 + 961904634169472/86029642954203a^7 + 464202423936059/86029642954203a^6 - 1282557053858260/86029642954203a^5 + 196943704554768/28676547651401a^4 + 132606232663362/28676547651401a^3 - 458458619477317/86029642954203a^2 + 25798828980231/28676547651401a + 25038610288890/28676547651401, 1501062567904/86029642954203a^17 - 771607303784/6617664842631a^16 + 15606769062190/86029642954203a^15 + 12703287136105/28676547651401a^14 - 164732765071253/86029642954203a^13 + 65185296219122/28676547651401a^12 + 52843821157892/86029642954203a^11 - 174088246478180/28676547651401a^10 + 310258984982784/28676547651401a^9 - 245384771343460/28676547651401a^8 - 140489231310788/86029642954203a^7 + 836260291526558/86029642954203a^6 - 739466483441962/86029642954203a^5 - 372905951531/86029642954203a^4 + 504251724689654/86029642954203a^3 - 275531828235359/86029642954203a^2 - 191930130488392/86029642954203*a + 118307333344817/86029642954203	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:	$13385.82469770262$	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 13385.82469770262 \cdot 1}{2\cdot\sqrt{19446011944726528000000}}\cr\approx \mathstrut & 0.188998800336351 \end{aligned}$

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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 + 11*x^16 + 6*x^15 - 60*x^14 + 122*x^13 - 141*x^12 + 74*x^11 + 54*x^10 - 122*x^9 + 71*x^8 + 36*x^7 - 70*x^6 + 16*x^5 + 33*x^4 - 10*x^3 - 11*x^2 - 4*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))))

Galois group

$C_6\times S_3$ (as 18T6):

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 36

The 18 conjugacy class representatives for

S_3 \times C_6

Character table for

S_3 \times C_6

Intermediate fields

\Q(\sqrt{7})

, 3.1.980.1,

\Q(\zeta_{7})^+

, 6.2.26891200.1,

\Q(\zeta_{28})^+

, 9.3.941192000.1

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

Galois closure:	data not computed
Degree 12 sibling:	12.0.368947264000000.7
Degree 18 sibling:	18.0.37980492079544000000000.2
Minimal sibling:	12.0.368947264000000.7

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.3.0.1}{3} }^{6}$	R	R	${\href{/padicField/11.6.0.1}{6} }^{3}$	${\href{/padicField/13.2.0.1}{2} }^{9}$	${\href{/padicField/17.6.0.1}{6} }^{3}$	${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.3.0.1}{3} }^{6}$	${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$	${\href{/padicField/41.6.0.1}{6} }^{3}$	${\href{/padicField/43.6.0.1}{6} }^{3}$	${\href{/padicField/47.3.0.1}{3} }^{6}$	${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$	${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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	2.3.2.6a1.1	$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$2$ 2	2.6.2.12a1.1	$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$	$2$	$6$	$12$	$C_6\times C_2$	$[2]^{6}$
$5$ 5	5.6.1.0a1.1	$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$5$ 5	5.6.2.6a1.2	$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$7$ 7	7.3.6.15a1.4	$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98736 x^{12} + 161280 x^{11} + 238464 x^{10} + 208640 x^{9} + 334080 x^{8} + 138240 x^{7} + 280320 x^{6} + 46080 x^{5} + 138240 x^{4} + 6144 x^{3} + 36871 x^{2} + 4117$	$6$	$3$	$15$	$C_6 \times C_3$	$[\ ]_{6}^{3}$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

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Fields

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Properties

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Normalized defining polynomial

Invariants

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

Class group and class number

Unit group

Class number formula

Galois group

Intermediate fields

Sibling fields

Frobenius cycle types

Local algebras for ramified primes

Spectrum of ring of integers

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	18.6.194...000.1

Degree	$18$
Signature	$[6, 6]$
Discriminant	$1.945\times 10^{22}$
Root discriminant	$17.31$
Ramified primes	$2,5,7$
Class number	$1$
Class group	trivial
Galois group	$S_3 \times C_6$ (as 18T6)

Properties

Related objects

Downloads

Learn more

Integral basis (with respect to field generator aaa)

Class group and class number

Sibling fields

Local algebras for ramified primes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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