LMFDB - Number field 18.0.450283905890997363000000000000.4

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225)

gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 132*y^15 + 234*y^14 - 198*y^13 - 42*y^12 + 36*y^11 + 1791*y^10 - 7439*y^9 + 17235*y^8 - 30852*y^7 + 45834*y^6 - 41490*y^5 - 1530*y^4 + 32460*y^3 - 9675*y^2 - 2925*y + 4225, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225)

$ x^{18} - 9 x^{17} + 45 x^{16} - 132 x^{15} + 234 x^{14} - 198 x^{13} - 42 x^{12} + 36 x^{11} + \cdots + 4225 $ x^18 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225

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:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-450283905890997363000000000000$ -450283905890997363000000000000 $\medspace = -\,2^{12}\cdot 3^{37}\cdot 5^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$44.40$	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:	$2^{2/3}3^{37/18}5^{2/3}\approx 44.40336798307826$
Ramified primes:	$2$, $3$, $5$ 2, 3, 5	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-3}) $
$\card{ \Gal(K/\Q) }$:	$18$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{36}a^{9}+\frac{1}{6}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}-\frac{1}{4}a-\frac{4}{9}$, $\frac{1}{72}a^{10}-\frac{1}{72}a^{9}-\frac{1}{8}a^{8}+\frac{1}{12}a^{7}+\frac{1}{24}a^{6}-\frac{1}{8}a^{5}+\frac{5}{24}a^{4}+\frac{1}{6}a^{3}-\frac{3}{8}a^{2}+\frac{29}{72}a-\frac{11}{72}$, $\frac{1}{72}a^{11}-\frac{1}{24}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{3}a^{5}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{36}a^{2}-\frac{3}{8}$, $\frac{1}{1872}a^{12}-\frac{1}{156}a^{11}+\frac{1}{1872}a^{10}+\frac{7}{936}a^{9}-\frac{5}{104}a^{8}+\frac{19}{78}a^{7}-\frac{49}{208}a^{6}+\frac{17}{52}a^{5}-\frac{119}{312}a^{4}-\frac{227}{936}a^{3}+\frac{139}{624}a^{2}+\frac{35}{117}a-\frac{5}{144}$, $\frac{1}{5616}a^{13}-\frac{1}{5616}a^{12}-\frac{1}{208}a^{11}-\frac{1}{5616}a^{10}-\frac{7}{2808}a^{9}+\frac{3}{52}a^{8}-\frac{133}{624}a^{7}-\frac{29}{624}a^{6}+\frac{3}{52}a^{5}+\frac{799}{2808}a^{4}-\frac{2705}{5616}a^{3}+\frac{51}{208}a^{2}-\frac{275}{5616}a-\frac{49}{432}$, $\frac{1}{5616}a^{14}-\frac{1}{5616}a^{12}+\frac{19}{2808}a^{11}+\frac{1}{468}a^{10}+\frac{4}{351}a^{9}-\frac{29}{624}a^{8}+\frac{3}{52}a^{7}-\frac{1}{39}a^{6}-\frac{137}{2808}a^{5}-\frac{1}{208}a^{4}+\frac{55}{1404}a^{3}+\frac{1909}{5616}a^{2}-\frac{55}{117}a+\frac{49}{216}$, $\frac{1}{196560}a^{15}+\frac{1}{196560}a^{14}+\frac{1}{19656}a^{13}-\frac{37}{196560}a^{12}+\frac{17}{28080}a^{11}+\frac{391}{98280}a^{10}-\frac{619}{65520}a^{9}-\frac{241}{21840}a^{8}+\frac{3919}{21840}a^{7}+\frac{13183}{98280}a^{6}-\frac{11177}{39312}a^{5}+\frac{21803}{196560}a^{4}-\frac{39643}{98280}a^{3}-\frac{3931}{39312}a^{2}+\frac{10349}{39312}a-\frac{23}{126}$, $\frac{1}{982800}a^{16}+\frac{1}{982800}a^{15}+\frac{1}{12285}a^{14}-\frac{1}{491400}a^{13}+\frac{17}{140400}a^{12}+\frac{3967}{982800}a^{11}-\frac{6407}{982800}a^{10}-\frac{4363}{327600}a^{9}-\frac{4271}{109200}a^{8}-\frac{51439}{982800}a^{7}-\frac{925}{39312}a^{6}-\frac{396797}{982800}a^{5}+\frac{191497}{491400}a^{4}-\frac{475}{19656}a^{3}-\frac{941}{15120}a^{2}+\frac{32311}{196560}a-\frac{7}{72}$, $\frac{1}{148804765200}a^{17}-\frac{62329}{148804765200}a^{16}-\frac{1819}{4960158840}a^{15}+\frac{1394}{1328613975}a^{14}+\frac{59}{7570450}a^{13}-\frac{22649813}{148804765200}a^{12}+\frac{20961919}{12400397100}a^{11}-\frac{909619169}{148804765200}a^{10}+\frac{1549519721}{148804765200}a^{9}+\frac{27638809}{5131198800}a^{8}-\frac{44927153}{212578236}a^{7}+\frac{6852511}{49601588400}a^{6}-\frac{28318617973}{74402382600}a^{5}-\frac{94312601}{236198040}a^{4}+\frac{1565183149}{7440238260}a^{3}+\frac{2059943677}{9920317680}a^{2}-\frac{2078291809}{5952190608}a-\frac{14458441}{57232602}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2*a - 1/2, 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/4*a^8 - 1/2*a^5 + 1/4*a^4 - 1/2*a^3 - 1/2*a^2 - 1/4, 1/36*a^9 + 1/6*a^6 + 1/4*a^5 - 1/2*a^4 + 1/6*a^3 - 1/4*a - 4/9, 1/72*a^10 - 1/72*a^9 - 1/8*a^8 + 1/12*a^7 + 1/24*a^6 - 1/8*a^5 + 5/24*a^4 + 1/6*a^3 - 3/8*a^2 + 29/72*a - 11/72, 1/72*a^11 - 1/24*a^8 + 1/8*a^7 - 1/4*a^6 + 1/3*a^5 - 1/8*a^4 - 3/8*a^3 + 1/36*a^2 - 3/8, 1/1872*a^12 - 1/156*a^11 + 1/1872*a^10 + 7/936*a^9 - 5/104*a^8 + 19/78*a^7 - 49/208*a^6 + 17/52*a^5 - 119/312*a^4 - 227/936*a^3 + 139/624*a^2 + 35/117*a - 5/144, 1/5616*a^13 - 1/5616*a^12 - 1/208*a^11 - 1/5616*a^10 - 7/2808*a^9 + 3/52*a^8 - 133/624*a^7 - 29/624*a^6 + 3/52*a^5 + 799/2808*a^4 - 2705/5616*a^3 + 51/208*a^2 - 275/5616*a - 49/432, 1/5616*a^14 - 1/5616*a^12 + 19/2808*a^11 + 1/468*a^10 + 4/351*a^9 - 29/624*a^8 + 3/52*a^7 - 1/39*a^6 - 137/2808*a^5 - 1/208*a^4 + 55/1404*a^3 + 1909/5616*a^2 - 55/117*a + 49/216, 1/196560*a^15 + 1/196560*a^14 + 1/19656*a^13 - 37/196560*a^12 + 17/28080*a^11 + 391/98280*a^10 - 619/65520*a^9 - 241/21840*a^8 + 3919/21840*a^7 + 13183/98280*a^6 - 11177/39312*a^5 + 21803/196560*a^4 - 39643/98280*a^3 - 3931/39312*a^2 + 10349/39312*a - 23/126, 1/982800*a^16 + 1/982800*a^15 + 1/12285*a^14 - 1/491400*a^13 + 17/140400*a^12 + 3967/982800*a^11 - 6407/982800*a^10 - 4363/327600*a^9 - 4271/109200*a^8 - 51439/982800*a^7 - 925/39312*a^6 - 396797/982800*a^5 + 191497/491400*a^4 - 475/19656*a^3 - 941/15120*a^2 + 32311/196560*a - 7/72, 1/148804765200*a^17 - 62329/148804765200*a^16 - 1819/4960158840*a^15 + 1394/1328613975*a^14 + 59/7570450*a^13 - 22649813/148804765200*a^12 + 20961919/12400397100*a^11 - 909619169/148804765200*a^10 + 1549519721/148804765200*a^9 + 27638809/5131198800*a^8 - 44927153/212578236*a^7 + 6852511/49601588400*a^6 - 28318617973/74402382600*a^5 - 94312601/236198040*a^4 + 1565183149/7440238260*a^3 + 2059943677/9920317680*a^2 - 2078291809/5952190608*a - 14458441/57232602

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $13$

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (assuming GRH)

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:	$8$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{250282}{9300297825} a^{17} - \frac{1520189}{9300297825} a^{16} + \frac{4653784}{9300297825} a^{15} + \frac{393376}{9300297825} a^{14} - \frac{129932}{28616301} a^{13} + \frac{27733556}{1860059565} a^{12} - \frac{200858576}{9300297825} a^{11} + \frac{31969864}{9300297825} a^{10} + \frac{437389606}{9300297825} a^{9} - \frac{20065646}{320699925} a^{8} - \frac{181120048}{1328613975} a^{7} + \frac{5721582416}{9300297825} a^{6} - \frac{2771723516}{1860059565} a^{5} + \frac{28044774536}{9300297825} a^{4} - \frac{1601341936}{372011913} a^{3} + \frac{838196936}{372011913} a^{2} + \frac{2619999154}{1860059565} a - \frac{1336363}{4088043} $ (250282)/(9300297825)a^(17) - (1520189)/(9300297825)a^(16) + (4653784)/(9300297825)a^(15) + (393376)/(9300297825)a^(14) - (129932)/(28616301)a^(13) + (27733556)/(1860059565)a^(12) - (200858576)/(9300297825)a^(11) + (31969864)/(9300297825)a^(10) + (437389606)/(9300297825)a^(9) - (20065646)/(320699925)a^(8) - (181120048)/(1328613975)a^(7) + (5721582416)/(9300297825)a^(6) - (2771723516)/(1860059565)a^(5) + (28044774536)/(9300297825)a^(4) - (1601341936)/(372011913)a^(3) + (838196936)/(372011913)a^(2) + (2619999154)/(1860059565)*a - (1336363)/(4088043) (order $6$)	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:	$\frac{16379}{57013320}a^{17}-\frac{442119}{190044400}a^{16}+\frac{2241503}{213799950}a^{15}-\frac{4612913}{171039960}a^{14}+\frac{2730863}{71266650}a^{13}-\frac{13011223}{570133200}a^{12}-\frac{1534472}{106899975}a^{11}-\frac{27441821}{570133200}a^{10}+\frac{424717157}{855199800}a^{9}-\frac{911008297}{570133200}a^{8}+\frac{151392349}{47511100}a^{7}-\frac{392389609}{68415984}a^{6}+\frac{1020436513}{122171400}a^{5}-\frac{721677157}{142533300}a^{4}-\frac{32202883}{57013320}a^{3}-\frac{25354991}{48868560}a^{2}+\frac{24858577}{28506660}a-\frac{166801}{187956}$, $\frac{3643477}{29760953040}a^{17}-\frac{181729}{165338628}a^{16}+\frac{11572357}{2125782360}a^{15}-\frac{156501053}{9920317680}a^{14}+\frac{15751151}{572326020}a^{13}-\frac{699129539}{29760953040}a^{12}-\frac{6941729}{14880476520}a^{11}-\frac{1813247}{265722795}a^{10}+\frac{421532317}{1984063536}a^{9}-\frac{172988155}{205247952}a^{8}+\frac{273008963}{137782190}a^{7}-\frac{14114200393}{3720119130}a^{6}+\frac{3712247249}{620019855}a^{5}-\frac{160359856613}{29760953040}a^{4}-\frac{3340215859}{2976095304}a^{3}+\frac{30163630685}{5952190608}a^{2}-\frac{6248955173}{5952190608}a-\frac{19651667}{10901448}$, $\frac{74207069}{148804765200}a^{17}-\frac{103484033}{24800794200}a^{16}+\frac{976227431}{49601588400}a^{15}-\frac{7795562633}{148804765200}a^{14}+\frac{11293678}{143081505}a^{13}-\frac{274325099}{7440238260}a^{12}-\frac{336329381}{5511287600}a^{11}-\frac{2595672467}{148804765200}a^{10}+\frac{16801187699}{18600595650}a^{9}-\frac{16063133497}{5131198800}a^{8}+\frac{315951370051}{49601588400}a^{7}-\frac{524956549151}{49601588400}a^{6}+\frac{438266824237}{29760953040}a^{5}-\frac{1370011842803}{148804765200}a^{4}-\frac{28466523019}{2976095304}a^{3}+\frac{1330555647}{110225752}a^{2}+\frac{72540724199}{14880476520}a-\frac{470349487}{457860816}$, $\frac{228337}{29760953040}a^{17}-\frac{103031}{2480079420}a^{16}+\frac{153067}{9920317680}a^{15}+\frac{247301}{327043440}a^{14}-\frac{341381}{81760860}a^{13}+\frac{145637729}{14880476520}a^{12}-\frac{121420609}{9920317680}a^{11}-\frac{12020441}{5952190608}a^{10}+\frac{68126881}{3720119130}a^{9}-\frac{1297463}{1026239760}a^{8}-\frac{279768199}{1417188240}a^{7}+\frac{4987413407}{9920317680}a^{6}-\frac{33037592639}{29760953040}a^{5}+\frac{611830319}{327043440}a^{4}-\frac{7424629417}{2976095304}a^{3}+\frac{46556498}{124003971}a^{2}+\frac{2710111519}{2976095304}a-\frac{68961763}{457860816}$, $\frac{348269}{885742650}a^{17}-\frac{522027253}{148804765200}a^{16}+\frac{99769115}{5952190608}a^{15}-\frac{6853258649}{148804765200}a^{14}+\frac{2559626347}{37201191300}a^{13}-\frac{4520448061}{148804765200}a^{12}-\frac{10193342119}{148804765200}a^{11}-\frac{18983647}{6200198550}a^{10}+\frac{112505891447}{148804765200}a^{9}-\frac{2378362927}{855199800}a^{8}+\frac{163758786173}{29760953040}a^{7}-\frac{337082647541}{37201191300}a^{6}+\frac{141787931951}{11446520400}a^{5}-\frac{6552772099}{850312944}a^{4}-\frac{162490899239}{14880476520}a^{3}+\frac{357800986757}{29760953040}a^{2}+\frac{2301633611}{283437648}a+\frac{217904275}{457860816}$, $\frac{13166581}{37201191300}a^{17}-\frac{1176529}{344455475}a^{16}+\frac{127470743}{7085941200}a^{15}-\frac{8557222273}{148804765200}a^{14}+\frac{247771793}{2125782360}a^{13}-\frac{275541257}{1984063536}a^{12}+\frac{3318482171}{49601588400}a^{11}-\frac{282834943}{10628911800}a^{10}+\frac{4651937857}{7085941200}a^{9}-\frac{15550800287}{5131198800}a^{8}+\frac{130049480857}{16533862800}a^{7}-\frac{386506508503}{24800794200}a^{6}+\frac{151983237325}{5952190608}a^{5}-\frac{4397906693783}{148804765200}a^{4}+\frac{83835975791}{4960158840}a^{3}+\frac{1915828945}{1984063536}a^{2}-\frac{134527727897}{29760953040}a+\frac{81674687}{38155068}$, $\frac{13619783}{148804765200}a^{17}-\frac{146281271}{74402382600}a^{16}+\frac{19120369}{1860059565}a^{15}-\frac{5275242811}{148804765200}a^{14}+\frac{513061772}{9300297825}a^{13}-\frac{5474340199}{148804765200}a^{12}-\frac{449607796}{9300297825}a^{11}+\frac{3004812049}{74402382600}a^{10}+\frac{9194015279}{21257823600}a^{9}-\frac{10164817183}{5131198800}a^{8}+\frac{8783855719}{2125782360}a^{7}-\frac{486074733373}{74402382600}a^{6}+\frac{211744077269}{18600595650}a^{5}-\frac{206433436267}{29760953040}a^{4}-\frac{23251885849}{2976095304}a^{3}+\frac{112127619623}{29760953040}a^{2}-\frac{1186508171}{5952190608}a-\frac{317147581}{228930408}$, $\frac{7846543}{49601588400}a^{17}-\frac{15117871}{12400397100}a^{16}+\frac{45020903}{8266931400}a^{15}-\frac{659680591}{49601588400}a^{14}+\frac{5577091}{317958900}a^{13}-\frac{60554189}{16533862800}a^{12}-\frac{111032759}{4960158840}a^{11}+\frac{3326129}{2755643800}a^{10}+\frac{11139364891}{49601588400}a^{9}-\frac{1300564961}{1710399600}a^{8}+\frac{9246840251}{6200198550}a^{7}-\frac{21894539911}{8266931400}a^{6}+\frac{85420296443}{24800794200}a^{5}-\frac{717007219}{423945200}a^{4}-\frac{801374845}{330677256}a^{3}+\frac{11658421279}{9920317680}a^{2}-\frac{545378539}{3306772560}a-\frac{8164721}{19077534}$ 16379/57013320a^17 - 442119/190044400a^16 + 2241503/213799950a^15 - 4612913/171039960a^14 + 2730863/71266650a^13 - 13011223/570133200a^12 - 1534472/106899975a^11 - 27441821/570133200a^10 + 424717157/855199800a^9 - 911008297/570133200a^8 + 151392349/47511100a^7 - 392389609/68415984a^6 + 1020436513/122171400a^5 - 721677157/142533300a^4 - 32202883/57013320a^3 - 25354991/48868560a^2 + 24858577/28506660a - 166801/187956, 3643477/29760953040a^17 - 181729/165338628a^16 + 11572357/2125782360a^15 - 156501053/9920317680a^14 + 15751151/572326020a^13 - 699129539/29760953040a^12 - 6941729/14880476520a^11 - 1813247/265722795a^10 + 421532317/1984063536a^9 - 172988155/205247952a^8 + 273008963/137782190a^7 - 14114200393/3720119130a^6 + 3712247249/620019855a^5 - 160359856613/29760953040a^4 - 3340215859/2976095304a^3 + 30163630685/5952190608a^2 - 6248955173/5952190608a - 19651667/10901448, 74207069/148804765200a^17 - 103484033/24800794200a^16 + 976227431/49601588400a^15 - 7795562633/148804765200a^14 + 11293678/143081505a^13 - 274325099/7440238260a^12 - 336329381/5511287600a^11 - 2595672467/148804765200a^10 + 16801187699/18600595650a^9 - 16063133497/5131198800a^8 + 315951370051/49601588400a^7 - 524956549151/49601588400a^6 + 438266824237/29760953040a^5 - 1370011842803/148804765200a^4 - 28466523019/2976095304a^3 + 1330555647/110225752a^2 + 72540724199/14880476520a - 470349487/457860816, 228337/29760953040a^17 - 103031/2480079420a^16 + 153067/9920317680a^15 + 247301/327043440a^14 - 341381/81760860a^13 + 145637729/14880476520a^12 - 121420609/9920317680a^11 - 12020441/5952190608a^10 + 68126881/3720119130a^9 - 1297463/1026239760a^8 - 279768199/1417188240a^7 + 4987413407/9920317680a^6 - 33037592639/29760953040a^5 + 611830319/327043440a^4 - 7424629417/2976095304a^3 + 46556498/124003971a^2 + 2710111519/2976095304a - 68961763/457860816, 348269/885742650a^17 - 522027253/148804765200a^16 + 99769115/5952190608a^15 - 6853258649/148804765200a^14 + 2559626347/37201191300a^13 - 4520448061/148804765200a^12 - 10193342119/148804765200a^11 - 18983647/6200198550a^10 + 112505891447/148804765200a^9 - 2378362927/855199800a^8 + 163758786173/29760953040a^7 - 337082647541/37201191300a^6 + 141787931951/11446520400a^5 - 6552772099/850312944a^4 - 162490899239/14880476520a^3 + 357800986757/29760953040a^2 + 2301633611/283437648a + 217904275/457860816, 13166581/37201191300a^17 - 1176529/344455475a^16 + 127470743/7085941200a^15 - 8557222273/148804765200a^14 + 247771793/2125782360a^13 - 275541257/1984063536a^12 + 3318482171/49601588400a^11 - 282834943/10628911800a^10 + 4651937857/7085941200a^9 - 15550800287/5131198800a^8 + 130049480857/16533862800a^7 - 386506508503/24800794200a^6 + 151983237325/5952190608a^5 - 4397906693783/148804765200a^4 + 83835975791/4960158840a^3 + 1915828945/1984063536a^2 - 134527727897/29760953040a + 81674687/38155068, 13619783/148804765200a^17 - 146281271/74402382600a^16 + 19120369/1860059565a^15 - 5275242811/148804765200a^14 + 513061772/9300297825a^13 - 5474340199/148804765200a^12 - 449607796/9300297825a^11 + 3004812049/74402382600a^10 + 9194015279/21257823600a^9 - 10164817183/5131198800a^8 + 8783855719/2125782360a^7 - 486074733373/74402382600a^6 + 211744077269/18600595650a^5 - 206433436267/29760953040a^4 - 23251885849/2976095304a^3 + 112127619623/29760953040a^2 - 1186508171/5952190608a - 317147581/228930408, 7846543/49601588400a^17 - 15117871/12400397100a^16 + 45020903/8266931400a^15 - 659680591/49601588400a^14 + 5577091/317958900a^13 - 60554189/16533862800a^12 - 111032759/4960158840a^11 + 3326129/2755643800a^10 + 11139364891/49601588400a^9 - 1300564961/1710399600a^8 + 9246840251/6200198550a^7 - 21894539911/8266931400a^6 + 85420296443/24800794200a^5 - 717007219/423945200a^4 - 801374845/330677256a^3 + 11658421279/9920317680a^2 - 545378539/3306772560a - 8164721/19077534 (assuming GRH)	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:	$ 20331620.6317 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 20331620.6317 \cdot 27}{6\cdot\sqrt{450283905890997363000000000000}}\cr\approx \mathstrut & 2.08094266387 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225)

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 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225, 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 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225);

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 - 9*x^17 + 45*x^16 - 132*x^15 + 234*x^14 - 198*x^13 - 42*x^12 + 36*x^11 + 1791*x^10 - 7439*x^9 + 17235*x^8 - 30852*x^7 + 45834*x^6 - 41490*x^5 - 1530*x^4 + 32460*x^3 - 9675*x^2 - 2925*x + 4225);

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

$C_3:S_3$ (as 18T4):

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 18

The 6 conjugacy class representatives for $C_3^2 : C_2$

Character table for $C_3^2 : C_2$

Intermediate fields

$\Q(\sqrt{-3}) $, 3.1.24300.1 x3, 3.1.972.1 x3, 3.1.24300.4 x3, 3.1.675.1 x3, 6.0.1771470000.3, 6.0.2834352.1, 6.0.1771470000.2, 6.0.1366875.1, 9.1.387420489000000.14 x9

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 9 sibling:	9.1.387420489000000.14
Minimal sibling:	9.1.387420489000000.14

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.3.0.1}{3} }^{6}$	${\href{/padicField/11.2.0.1}{2} }^{9}$	${\href{/padicField/13.1.0.1}{1} }^{18}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.2.0.1}{2} }^{9}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.3.0.1}{3} }^{6}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.2.0.1}{2} }^{9}$

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	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$3$ 3		Deg $18$	$18$	$1$	$37$
$5$ 5	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$

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