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Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225)

gp:K = bnfinit(y^18 - 2*y^17 + 17*y^15 - 17*y^14 + 51*y^13 + 119*y^12 - 476*y^11 + 493*y^10 + 1632*y^9 - 3638*y^8 - 3502*y^7 + 6936*y^6 + 1972*y^5 - 3451*y^4 + 3264*y^3 + 17*y^2 - 750*y + 225, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225)

$ x^{18} - 2 x^{17} + 17 x^{15} - 17 x^{14} + 51 x^{13} + 119 x^{12} - 476 x^{11} + 493 x^{10} + \cdots + 225 $ x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$18$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$(2, 8)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$4768875488962616064464333777$ 4768875488962616064464333777 $\medspace = 7^{8}\cdot 17^{17}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$34.49$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$7^{1/2}17^{17/18}\approx 38.427282044851594$
Ramified primes:	$7$, $17$ 7, 17	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q(\sqrt{17}) $
$\Aut(K/\Q)$:	$C_2$	comment:Automorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphism_group(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{15}a^{9}+\frac{1}{15}a^{8}-\frac{2}{15}a^{7}+\frac{7}{15}a^{6}+\frac{1}{15}a^{5}-\frac{1}{3}a^{4}+\frac{7}{15}a^{3}+\frac{4}{15}a^{2}+\frac{1}{5}a$, $\frac{1}{15}a^{10}+\frac{2}{15}a^{8}-\frac{1}{15}a^{7}-\frac{1}{15}a^{6}-\frac{1}{15}a^{5}+\frac{2}{15}a^{4}+\frac{2}{15}a^{3}+\frac{4}{15}a^{2}+\frac{2}{15}a$, $\frac{1}{15}a^{11}+\frac{2}{15}a^{8}-\frac{7}{15}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{2}{15}a^{4}-\frac{1}{3}a^{3}-\frac{1}{15}a^{2}-\frac{1}{15}a$, $\frac{1}{195}a^{12}+\frac{4}{195}a^{11}+\frac{1}{39}a^{10}+\frac{4}{195}a^{9}-\frac{4}{65}a^{8}+\frac{11}{65}a^{7}-\frac{17}{65}a^{6}+\frac{18}{65}a^{5}+\frac{38}{195}a^{4}+\frac{8}{195}a^{3}-\frac{62}{195}a^{2}-\frac{73}{195}a-\frac{3}{13}$, $\frac{1}{195}a^{13}+\frac{2}{195}a^{11}-\frac{1}{65}a^{10}-\frac{2}{195}a^{9}+\frac{29}{195}a^{8}-\frac{79}{195}a^{7}-\frac{28}{195}a^{6}-\frac{7}{39}a^{5}+\frac{38}{195}a^{4}-\frac{27}{65}a^{3}-\frac{7}{195}a^{2}+\frac{1}{15}a-\frac{1}{13}$, $\frac{1}{975}a^{14}-\frac{1}{975}a^{12}-\frac{1}{65}a^{11}-\frac{2}{65}a^{10}-\frac{3}{325}a^{9}-\frac{32}{195}a^{8}+\frac{68}{975}a^{7}+\frac{79}{975}a^{6}-\frac{397}{975}a^{5}-\frac{4}{25}a^{4}+\frac{476}{975}a^{3}-\frac{217}{975}a^{2}+\frac{7}{195}a-\frac{6}{13}$, $\frac{1}{4875}a^{15}+\frac{2}{4875}a^{14}-\frac{1}{4875}a^{13}+\frac{1}{1625}a^{12}+\frac{4}{975}a^{11}+\frac{32}{1625}a^{10}+\frac{97}{4875}a^{9}-\frac{167}{4875}a^{8}-\frac{102}{325}a^{7}+\frac{41}{4875}a^{6}+\frac{34}{75}a^{5}-\frac{896}{4875}a^{4}+\frac{283}{975}a^{3}+\frac{376}{4875}a^{2}-\frac{28}{65}a-\frac{11}{65}$, $\frac{1}{102375}a^{16}-\frac{1}{14625}a^{15}+\frac{2}{34125}a^{14}-\frac{188}{102375}a^{13}-\frac{1}{14625}a^{12}+\frac{397}{34125}a^{11}-\frac{467}{102375}a^{10}+\frac{562}{20475}a^{9}+\frac{103}{1125}a^{8}+\frac{199}{2625}a^{7}+\frac{341}{102375}a^{6}+\frac{21514}{102375}a^{5}+\frac{1249}{4875}a^{4}-\frac{25009}{102375}a^{3}+\frac{18841}{102375}a^{2}-\frac{179}{1365}a+\frac{6}{35}$, $\frac{1}{313358232607875}a^{17}+\frac{1404856129}{313358232607875}a^{16}+\frac{25409243719}{313358232607875}a^{15}+\frac{155108268703}{313358232607875}a^{14}+\frac{7330690499}{4820895886275}a^{13}-\frac{209454488186}{313358232607875}a^{12}+\frac{41219692024}{3443497061625}a^{11}-\frac{2913813250412}{313358232607875}a^{10}-\frac{739245317434}{24104479431375}a^{9}-\frac{1195848720691}{313358232607875}a^{8}+\frac{61169396576297}{313358232607875}a^{7}-\frac{3361738081379}{8953092360225}a^{6}+\frac{186254065181}{777563852625}a^{5}+\frac{4692836456396}{12534329304315}a^{4}+\frac{11845255351616}{44765461801125}a^{3}+\frac{6362826393968}{44765461801125}a^{2}+\frac{39793089398}{134777734455}a+\frac{351822543028}{1392703256035}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 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*a^2 + 1/3*a, 1/15*a^9 + 1/15*a^8 - 2/15*a^7 + 7/15*a^6 + 1/15*a^5 - 1/3*a^4 + 7/15*a^3 + 4/15*a^2 + 1/5*a, 1/15*a^10 + 2/15*a^8 - 1/15*a^7 - 1/15*a^6 - 1/15*a^5 + 2/15*a^4 + 2/15*a^3 + 4/15*a^2 + 2/15*a, 1/15*a^11 + 2/15*a^8 - 7/15*a^7 + 1/3*a^6 + 1/3*a^5 + 2/15*a^4 - 1/3*a^3 - 1/15*a^2 - 1/15*a, 1/195*a^12 + 4/195*a^11 + 1/39*a^10 + 4/195*a^9 - 4/65*a^8 + 11/65*a^7 - 17/65*a^6 + 18/65*a^5 + 38/195*a^4 + 8/195*a^3 - 62/195*a^2 - 73/195*a - 3/13, 1/195*a^13 + 2/195*a^11 - 1/65*a^10 - 2/195*a^9 + 29/195*a^8 - 79/195*a^7 - 28/195*a^6 - 7/39*a^5 + 38/195*a^4 - 27/65*a^3 - 7/195*a^2 + 1/15*a - 1/13, 1/975*a^14 - 1/975*a^12 - 1/65*a^11 - 2/65*a^10 - 3/325*a^9 - 32/195*a^8 + 68/975*a^7 + 79/975*a^6 - 397/975*a^5 - 4/25*a^4 + 476/975*a^3 - 217/975*a^2 + 7/195*a - 6/13, 1/4875*a^15 + 2/4875*a^14 - 1/4875*a^13 + 1/1625*a^12 + 4/975*a^11 + 32/1625*a^10 + 97/4875*a^9 - 167/4875*a^8 - 102/325*a^7 + 41/4875*a^6 + 34/75*a^5 - 896/4875*a^4 + 283/975*a^3 + 376/4875*a^2 - 28/65*a - 11/65, 1/102375*a^16 - 1/14625*a^15 + 2/34125*a^14 - 188/102375*a^13 - 1/14625*a^12 + 397/34125*a^11 - 467/102375*a^10 + 562/20475*a^9 + 103/1125*a^8 + 199/2625*a^7 + 341/102375*a^6 + 21514/102375*a^5 + 1249/4875*a^4 - 25009/102375*a^3 + 18841/102375*a^2 - 179/1365*a + 6/35, 1/313358232607875*a^17 + 1404856129/313358232607875*a^16 + 25409243719/313358232607875*a^15 + 155108268703/313358232607875*a^14 + 7330690499/4820895886275*a^13 - 209454488186/313358232607875*a^12 + 41219692024/3443497061625*a^11 - 2913813250412/313358232607875*a^10 - 739245317434/24104479431375*a^9 - 1195848720691/313358232607875*a^8 + 61169396576297/313358232607875*a^7 - 3361738081379/8953092360225*a^6 + 186254065181/777563852625*a^5 + 4692836456396/12534329304315*a^4 + 11845255351616/44765461801125*a^3 + 6362826393968/44765461801125*a^2 + 39793089398/134777734455*a + 351822543028/1392703256035

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$

Class group and class number

Ideal class group:		Trivial group, which has order $1$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

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

oscar:UK, fUK = unit_group(OK)

Rank:	$9$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{267491154221}{313358232607875}a^{17}-\frac{554320101008}{313358232607875}a^{16}+\frac{3024157772}{8034826477125}a^{15}+\frac{4410102035996}{313358232607875}a^{14}-\frac{4945784171834}{313358232607875}a^{13}+\frac{5221212374426}{104452744202625}a^{12}+\frac{28878812742842}{313358232607875}a^{11}-\frac{124888144421648}{313358232607875}a^{10}+\frac{154619313560258}{313358232607875}a^{9}+\frac{127173194556781}{104452744202625}a^{8}-\frac{189554544285202}{62671646521575}a^{7}-\frac{728864082056752}{313358232607875}a^{6}+\frac{3339024985532}{673888672275}a^{5}+\frac{298299557428762}{313358232607875}a^{4}-\frac{96387412811423}{62671646521575}a^{3}+\frac{166702789556143}{104452744202625}a^{2}+\frac{869702132}{10367518035}a+\frac{105352959437}{1392703256035}$, $\frac{26852169814}{104452744202625}a^{17}-\frac{748335121}{1606965295425}a^{16}+\frac{89054677}{278540651207}a^{15}+\frac{16642535171}{4178109768105}a^{14}-\frac{418639133749}{104452744202625}a^{13}+\frac{637637760023}{34817581400875}a^{12}+\frac{702411857246}{20890548840525}a^{11}-\frac{9779770502759}{104452744202625}a^{10}+\frac{16619469548726}{104452744202625}a^{9}+\frac{11674219472038}{34817581400875}a^{8}-\frac{86821478330378}{104452744202625}a^{7}-\frac{44792413425889}{104452744202625}a^{6}+\frac{4212733718693}{3369443361375}a^{5}-\frac{186239610105071}{104452744202625}a^{4}-\frac{47853906475451}{34817581400875}a^{3}+\frac{29870585912208}{6963516280175}a^{2}+\frac{377435310754}{134777734455}a-\frac{235949549816}{278540651207}$, $\frac{1866495444}{2678275492375}a^{17}-\frac{3525882872}{1790618472045}a^{16}+\frac{542908405316}{313358232607875}a^{15}+\frac{983651001859}{104452744202625}a^{14}-\frac{58769033324}{3443497061625}a^{13}+\frac{15238950907631}{313358232607875}a^{12}+\frac{209746370668}{6963516280175}a^{11}-\frac{100905850593602}{313358232607875}a^{10}+\frac{24436000630532}{44765461801125}a^{9}+\frac{199130168637121}{313358232607875}a^{8}-\frac{90526719702429}{34817581400875}a^{7}-\frac{327450673007722}{313358232607875}a^{6}+\frac{7429546108663}{1444047154875}a^{5}-\frac{3938191553056}{104452744202625}a^{4}-\frac{980451923850331}{313358232607875}a^{3}+\frac{431440007053681}{313358232607875}a^{2}-\frac{1540681117}{10367518035}a-\frac{19766344656}{198957608005}$, $\frac{6514546781}{2984364120075}a^{17}-\frac{1126296696743}{313358232607875}a^{16}-\frac{73310786482}{44765461801125}a^{15}+\frac{1307923556048}{34817581400875}a^{14}-\frac{7572254365271}{313358232607875}a^{13}+\frac{333735797726}{3443497061625}a^{12}+\frac{31753914513379}{104452744202625}a^{11}-\frac{297537006932699}{313358232607875}a^{10}+\frac{45288040582741}{62671646521575}a^{9}+\frac{13869045314626}{3443497061625}a^{8}-\frac{702547048953271}{104452744202625}a^{7}-\frac{32\cdots 33}{313358232607875}a^{6}+\frac{132963479145013}{10108330084125}a^{5}+\frac{154912548914183}{14921820600375}a^{4}-\frac{23\cdots 58}{313358232607875}a^{3}+\frac{833075483657707}{313358232607875}a^{2}+\frac{14369499002}{3455839345}a-\frac{1436948785784}{1392703256035}$, $\frac{22300096991}{313358232607875}a^{17}-\frac{4103331968}{6963516280175}a^{16}+\frac{45324708523}{313358232607875}a^{15}+\frac{64260550988}{44765461801125}a^{14}-\frac{820747973018}{104452744202625}a^{13}+\frac{4546157537}{62671646521575}a^{12}-\frac{1354891300244}{62671646521575}a^{11}-\frac{1033442734417}{8034826477125}a^{10}+\frac{2012243428718}{20890548840525}a^{9}+\frac{279373679219}{12534329304315}a^{8}-\frac{317709990577037}{313358232607875}a^{7}+\frac{3691771922719}{104452744202625}a^{6}+\frac{25075891523057}{10108330084125}a^{5}+\frac{512052411364348}{313358232607875}a^{4}+\frac{50683877719487}{34817581400875}a^{3}+\frac{181603293265988}{313358232607875}a^{2}-\frac{195260071264}{134777734455}a+\frac{663259592139}{1392703256035}$, $\frac{198283711414}{44765461801125}a^{17}-\frac{210911305973}{24104479431375}a^{16}+\frac{46614206689}{44765461801125}a^{15}+\frac{22523873661376}{313358232607875}a^{14}-\frac{22277236822361}{313358232607875}a^{13}+\frac{10949302075843}{44765461801125}a^{12}+\frac{154317872711356}{313358232607875}a^{11}-\frac{24989153950283}{12534329304315}a^{10}+\frac{690004447393007}{313358232607875}a^{9}+\frac{295597386960308}{44765461801125}a^{8}-\frac{184601280432067}{12534329304315}a^{7}-\frac{46\cdots 47}{313358232607875}a^{6}+\frac{50441064043459}{2021666016825}a^{5}+\frac{446925952820636}{44765461801125}a^{4}-\frac{546668165400368}{62671646521575}a^{3}+\frac{38\cdots 06}{313358232607875}a^{2}+\frac{3656734240}{26955546891}a-\frac{1336014646732}{1392703256035}$, $\frac{102611790773}{44765461801125}a^{17}-\frac{1247721140242}{313358232607875}a^{16}-\frac{91490586712}{44765461801125}a^{15}+\frac{12699506910422}{313358232607875}a^{14}-\frac{10021285421752}{313358232607875}a^{13}+\frac{4467451244768}{44765461801125}a^{12}+\frac{96227579762948}{313358232607875}a^{11}-\frac{5326238868821}{4820895886275}a^{10}+\frac{249255361292383}{313358232607875}a^{9}+\frac{183998804962573}{44765461801125}a^{8}-\frac{24\cdots 44}{313358232607875}a^{7}-\frac{33\cdots 56}{313358232607875}a^{6}+\frac{140061372032509}{10108330084125}a^{5}+\frac{434246666277283}{44765461801125}a^{4}-\frac{15\cdots 24}{313358232607875}a^{3}+\frac{67748038027324}{12534329304315}a^{2}+\frac{54787259903}{26955546891}a-\frac{443067397299}{278540651207}$, $\frac{29865326492}{10108330084125}a^{17}-\frac{50487910931}{10108330084125}a^{16}-\frac{86919057}{44925911485}a^{15}+\frac{101940060766}{2021666016825}a^{14}-\frac{353403781727}{10108330084125}a^{13}+\frac{151091910442}{1123147787125}a^{12}+\frac{4028475801524}{10108330084125}a^{11}-\frac{13213858696307}{10108330084125}a^{10}+\frac{10348092343874}{10108330084125}a^{9}+\frac{5938044268022}{1123147787125}a^{8}-\frac{18762389384314}{2021666016825}a^{7}-\frac{5526023087077}{404333203365}a^{6}+\frac{2325541038199}{134777734455}a^{5}+\frac{25466579290382}{2021666016825}a^{4}-\frac{15069981369218}{2021666016825}a^{3}+\frac{17143023523844}{3369443361375}a^{2}+\frac{17074976701}{10367518035}a-\frac{73342427014}{44925911485}$, $\frac{34321461198}{34817581400875}a^{17}-\frac{27853879909}{24104479431375}a^{16}-\frac{60855917056}{62671646521575}a^{15}+\frac{327834164102}{20890548840525}a^{14}-\frac{1015314300817}{313358232607875}a^{13}+\frac{14716490583094}{313358232607875}a^{12}+\frac{758858950391}{4973940200125}a^{11}-\frac{106521876312907}{313358232607875}a^{10}+\frac{2268943877611}{12534329304315}a^{9}+\frac{545068490935229}{313358232607875}a^{8}-\frac{71719524278144}{34817581400875}a^{7}-\frac{238785809927894}{44765461801125}a^{6}+\frac{22400248237166}{10108330084125}a^{5}+\frac{460437465330616}{104452744202625}a^{4}+\frac{38985447426997}{44765461801125}a^{3}+\frac{146646828663527}{44765461801125}a^{2}+\frac{302754219412}{134777734455}a+\frac{1198067645357}{1392703256035}$ 267491154221/313358232607875a^17 - 554320101008/313358232607875a^16 + 3024157772/8034826477125a^15 + 4410102035996/313358232607875a^14 - 4945784171834/313358232607875a^13 + 5221212374426/104452744202625a^12 + 28878812742842/313358232607875a^11 - 124888144421648/313358232607875a^10 + 154619313560258/313358232607875a^9 + 127173194556781/104452744202625a^8 - 189554544285202/62671646521575a^7 - 728864082056752/313358232607875a^6 + 3339024985532/673888672275a^5 + 298299557428762/313358232607875a^4 - 96387412811423/62671646521575a^3 + 166702789556143/104452744202625a^2 + 869702132/10367518035a + 105352959437/1392703256035, 26852169814/104452744202625a^17 - 748335121/1606965295425a^16 + 89054677/278540651207a^15 + 16642535171/4178109768105a^14 - 418639133749/104452744202625a^13 + 637637760023/34817581400875a^12 + 702411857246/20890548840525a^11 - 9779770502759/104452744202625a^10 + 16619469548726/104452744202625a^9 + 11674219472038/34817581400875a^8 - 86821478330378/104452744202625a^7 - 44792413425889/104452744202625a^6 + 4212733718693/3369443361375a^5 - 186239610105071/104452744202625a^4 - 47853906475451/34817581400875a^3 + 29870585912208/6963516280175a^2 + 377435310754/134777734455a - 235949549816/278540651207, 1866495444/2678275492375a^17 - 3525882872/1790618472045a^16 + 542908405316/313358232607875a^15 + 983651001859/104452744202625a^14 - 58769033324/3443497061625a^13 + 15238950907631/313358232607875a^12 + 209746370668/6963516280175a^11 - 100905850593602/313358232607875a^10 + 24436000630532/44765461801125a^9 + 199130168637121/313358232607875a^8 - 90526719702429/34817581400875a^7 - 327450673007722/313358232607875a^6 + 7429546108663/1444047154875a^5 - 3938191553056/104452744202625a^4 - 980451923850331/313358232607875a^3 + 431440007053681/313358232607875a^2 - 1540681117/10367518035a - 19766344656/198957608005, 6514546781/2984364120075a^17 - 1126296696743/313358232607875a^16 - 73310786482/44765461801125a^15 + 1307923556048/34817581400875a^14 - 7572254365271/313358232607875a^13 + 333735797726/3443497061625a^12 + 31753914513379/104452744202625a^11 - 297537006932699/313358232607875a^10 + 45288040582741/62671646521575a^9 + 13869045314626/3443497061625a^8 - 702547048953271/104452744202625a^7 - 3282448997955433/313358232607875a^6 + 132963479145013/10108330084125a^5 + 154912548914183/14921820600375a^4 - 2366549555916058/313358232607875a^3 + 833075483657707/313358232607875a^2 + 14369499002/3455839345a - 1436948785784/1392703256035, 22300096991/313358232607875a^17 - 4103331968/6963516280175a^16 + 45324708523/313358232607875a^15 + 64260550988/44765461801125a^14 - 820747973018/104452744202625a^13 + 4546157537/62671646521575a^12 - 1354891300244/62671646521575a^11 - 1033442734417/8034826477125a^10 + 2012243428718/20890548840525a^9 + 279373679219/12534329304315a^8 - 317709990577037/313358232607875a^7 + 3691771922719/104452744202625a^6 + 25075891523057/10108330084125a^5 + 512052411364348/313358232607875a^4 + 50683877719487/34817581400875a^3 + 181603293265988/313358232607875a^2 - 195260071264/134777734455a + 663259592139/1392703256035, 198283711414/44765461801125a^17 - 210911305973/24104479431375a^16 + 46614206689/44765461801125a^15 + 22523873661376/313358232607875a^14 - 22277236822361/313358232607875a^13 + 10949302075843/44765461801125a^12 + 154317872711356/313358232607875a^11 - 24989153950283/12534329304315a^10 + 690004447393007/313358232607875a^9 + 295597386960308/44765461801125a^8 - 184601280432067/12534329304315a^7 - 4677126341719147/313358232607875a^6 + 50441064043459/2021666016825a^5 + 446925952820636/44765461801125a^4 - 546668165400368/62671646521575a^3 + 3835362135627206/313358232607875a^2 + 3656734240/26955546891a - 1336014646732/1392703256035, 102611790773/44765461801125a^17 - 1247721140242/313358232607875a^16 - 91490586712/44765461801125a^15 + 12699506910422/313358232607875a^14 - 10021285421752/313358232607875a^13 + 4467451244768/44765461801125a^12 + 96227579762948/313358232607875a^11 - 5326238868821/4820895886275a^10 + 249255361292383/313358232607875a^9 + 183998804962573/44765461801125a^8 - 2493901373706044/313358232607875a^7 - 3311619364724756/313358232607875a^6 + 140061372032509/10108330084125a^5 + 434246666277283/44765461801125a^4 - 1510498143777524/313358232607875a^3 + 67748038027324/12534329304315a^2 + 54787259903/26955546891a - 443067397299/278540651207, 29865326492/10108330084125a^17 - 50487910931/10108330084125a^16 - 86919057/44925911485a^15 + 101940060766/2021666016825a^14 - 353403781727/10108330084125a^13 + 151091910442/1123147787125a^12 + 4028475801524/10108330084125a^11 - 13213858696307/10108330084125a^10 + 10348092343874/10108330084125a^9 + 5938044268022/1123147787125a^8 - 18762389384314/2021666016825a^7 - 5526023087077/404333203365a^6 + 2325541038199/134777734455a^5 + 25466579290382/2021666016825a^4 - 15069981369218/2021666016825a^3 + 17143023523844/3369443361375a^2 + 17074976701/10367518035a - 73342427014/44925911485, 34321461198/34817581400875a^17 - 27853879909/24104479431375a^16 - 60855917056/62671646521575a^15 + 327834164102/20890548840525a^14 - 1015314300817/313358232607875a^13 + 14716490583094/313358232607875a^12 + 758858950391/4973940200125a^11 - 106521876312907/313358232607875a^10 + 2268943877611/12534329304315a^9 + 545068490935229/313358232607875a^8 - 71719524278144/34817581400875a^7 - 238785809927894/44765461801125a^6 + 22400248237166/10108330084125a^5 + 460437465330616/104452744202625a^4 + 38985447426997/44765461801125a^3 + 146646828663527/44765461801125a^2 + 302754219412/134777734455*a + 1198067645357/1392703256035	comment:Fundamental units 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:	$ 10997947.525 $	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)
Unit signature rank:	$ 2 $

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 10997947.525 \cdot 1}{2\cdot\sqrt{4768875488962616064464333777}}\cr\approx \mathstrut & 0.77369999480 \end{aligned}\]

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 2*x^17 + 17*x^15 - 17*x^14 + 51*x^13 + 119*x^12 - 476*x^11 + 493*x^10 + 1632*x^9 - 3638*x^8 - 3502*x^7 + 6936*x^6 + 1972*x^5 - 3451*x^4 + 3264*x^3 + 17*x^2 - 750*x + 225); 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

$D_{18}$ (as 18T13):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A solvable group of order 36

The 12 conjugacy class representatives for $D_{18}$

Character table for $D_{18}$

Intermediate fields

$\Q(\sqrt{17}) $, 3.1.2023.1, 6.2.69572993.1, 9.1.16748793615841.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Galois closure:	data not computed
Degree 18 sibling:	18.0.33382128422738312451250336439.1
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.9.0.1}{9} }^{2}$	${\href{/padicField/3.2.0.1}{2} }^{9}$	${\href{/padicField/5.2.0.1}{2} }^{9}$	R	${\href{/padicField/11.6.0.1}{6} }^{3}$	${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	R	${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	$18$	$18$	${\href{/padicField/31.2.0.1}{2} }^{9}$	$18$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.9.0.1}{9} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.9.0.1}{9} }^{2}$	${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
$7$ 7	7.2.1.0a1.1	$x^{2} + 6 x + 3$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	7.2.2.2a1.2	$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	7.2.2.2a1.2	$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	7.2.2.2a1.2	$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	7.2.2.2a1.2	$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
$17$ 17	17.1.18.17a1.1	$x^{18} + 17$	$18$	$1$	$17$	$D_{18}$	$$[\ ]_{18}^{2}$$

Spectrum of ring of integers

Additional information

This field is associated with the 17-torsion points on the elliptic curves from the isogeny class 49.a1, a curve with complex multiplication by $\Q(\sqrt{-7})$.

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