LMFDB - Number field 18.0.242190571378050467501912064.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729)

gp: K = bnfinit(y^18 - 2*y^17 - 4*y^16 + 22*y^15 + 44*y^14 - 74*y^13 + 223*y^12 + 312*y^11 - 1620*y^10 - 576*y^9 + 5508*y^8 - 2916*y^7 - 3861*y^6 + 7614*y^5 + 2916*y^4 - 486*y^3 + 2916*y^2 + 1458*y + 729, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729)

$ x^{18} - 2 x^{17} - 4 x^{16} + 22 x^{15} + 44 x^{14} - 74 x^{13} + 223 x^{12} + 312 x^{11} - 1620 x^{10} + \cdots + 729 $ x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729

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:	$-242190571378050467501912064$ -242190571378050467501912064 $\medspace = -\,2^{12}\cdot 3^{9}\cdot 113^{9}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$29.23$	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:	$2^{2/3}3^{1/2}113^{1/2}\approx 29.22715298876584$
Ramified primes:	$2$, $3$, $113$ 2, 3, 113	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-339}) $
$\card{ \Gal(K/\Q) }$:	$18$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{18}a^{9}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{18}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{54}a^{10}+\frac{1}{54}a^{9}+\frac{1}{27}a^{8}-\frac{5}{54}a^{7}-\frac{1}{54}a^{6}-\frac{10}{27}a^{5}-\frac{11}{54}a^{4}+\frac{7}{18}a^{3}-\frac{1}{9}a^{2}+\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{54}a^{11}+\frac{1}{54}a^{9}-\frac{1}{54}a^{8}-\frac{1}{27}a^{7}+\frac{5}{54}a^{6}+\frac{7}{18}a^{5}+\frac{10}{27}a^{4}+\frac{7}{18}a^{3}-\frac{7}{18}a^{2}+\frac{1}{3}a-\frac{1}{6}$, $\frac{1}{486}a^{12}+\frac{1}{486}a^{11}-\frac{1}{486}a^{10}+\frac{1}{486}a^{9}-\frac{25}{486}a^{8}+\frac{31}{486}a^{7}+\frac{55}{486}a^{6}+\frac{19}{54}a^{5}+\frac{7}{54}a^{4}-\frac{7}{18}a^{3}+\frac{1}{6}a^{2}-\frac{7}{18}a+\frac{4}{9}$, $\frac{1}{972}a^{13}+\frac{7}{972}a^{11}-\frac{7}{972}a^{10}+\frac{1}{972}a^{9}-\frac{25}{972}a^{8}-\frac{19}{324}a^{7}-\frac{127}{972}a^{6}+\frac{17}{108}a^{5}+\frac{11}{36}a^{4}+\frac{5}{36}a^{3}-\frac{5}{12}a^{2}-\frac{1}{3}a-\frac{5}{36}$, $\frac{1}{2916}a^{14}+\frac{1}{2916}a^{13}-\frac{1}{2916}a^{12}-\frac{13}{1458}a^{11}+\frac{1}{1458}a^{10}+\frac{1}{729}a^{9}-\frac{20}{729}a^{8}-\frac{5}{81}a^{7}-\frac{13}{162}a^{6}-\frac{13}{27}a^{5}+\frac{2}{9}a^{4}+\frac{19}{54}a^{3}-\frac{37}{108}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{8748}a^{15}+\frac{1}{8748}a^{14}-\frac{1}{2187}a^{13}-\frac{1}{4374}a^{12}-\frac{49}{8748}a^{11}+\frac{55}{8748}a^{10}-\frac{221}{8748}a^{9}-\frac{73}{2916}a^{8}-\frac{259}{2916}a^{7}-\frac{43}{324}a^{6}-\frac{79}{324}a^{5}-\frac{61}{324}a^{4}-\frac{23}{162}a^{3}-\frac{25}{54}a^{2}-\frac{53}{108}a-\frac{5}{12}$, $\frac{1}{7479540}a^{16}+\frac{83}{3739770}a^{15}-\frac{101}{3739770}a^{14}-\frac{112}{373977}a^{13}+\frac{734}{1869885}a^{12}+\frac{37489}{7479540}a^{11}+\frac{64549}{7479540}a^{10}-\frac{8669}{498636}a^{9}+\frac{93007}{2493180}a^{8}-\frac{29729}{277020}a^{7}+\frac{104137}{831060}a^{6}-\frac{12727}{277020}a^{5}-\frac{5047}{69255}a^{4}-\frac{3799}{18468}a^{3}+\frac{38807}{92340}a^{2}-\frac{773}{2052}a-\frac{14197}{30780}$, $\frac{1}{23096108963700}a^{17}+\frac{1011097}{23096108963700}a^{16}-\frac{174325913}{11548054481850}a^{15}-\frac{240962477}{23096108963700}a^{14}+\frac{944156173}{11548054481850}a^{13}-\frac{1177493041}{1154805448185}a^{12}+\frac{76226785693}{23096108963700}a^{11}-\frac{53824542227}{7698702987900}a^{10}+\frac{170559783637}{7698702987900}a^{9}-\frac{22217510593}{2566234329300}a^{8}+\frac{422992709}{27012992940}a^{7}+\frac{126909456157}{855411443100}a^{6}-\frac{7917594305}{17108228862}a^{5}-\frac{74027753}{142568573850}a^{4}+\frac{9236862767}{285137147700}a^{3}-\frac{5252026529}{23761428975}a^{2}+\frac{3099029453}{95045715900}a+\frac{4111309651}{31681905300}$ 1, a, a^2, a^3, a^4, a^5, 1/3*a^6 + 1/3*a^4 + 1/3*a^2, 1/3*a^7 + 1/3*a^5 + 1/3*a^3, 1/9*a^8 - 1/9*a^7 + 1/9*a^6 + 2/9*a^5 + 4/9*a^4 - 1/9*a^3 + 1/3*a, 1/18*a^9 - 1/6*a^6 + 1/3*a^5 + 1/3*a^4 - 1/18*a^3 + 1/3*a^2 - 1/3*a - 1/2, 1/54*a^10 + 1/54*a^9 + 1/27*a^8 - 5/54*a^7 - 1/54*a^6 - 10/27*a^5 - 11/54*a^4 + 7/18*a^3 - 1/9*a^2 + 1/6*a + 1/6, 1/54*a^11 + 1/54*a^9 - 1/54*a^8 - 1/27*a^7 + 5/54*a^6 + 7/18*a^5 + 10/27*a^4 + 7/18*a^3 - 7/18*a^2 + 1/3*a - 1/6, 1/486*a^12 + 1/486*a^11 - 1/486*a^10 + 1/486*a^9 - 25/486*a^8 + 31/486*a^7 + 55/486*a^6 + 19/54*a^5 + 7/54*a^4 - 7/18*a^3 + 1/6*a^2 - 7/18*a + 4/9, 1/972*a^13 + 7/972*a^11 - 7/972*a^10 + 1/972*a^9 - 25/972*a^8 - 19/324*a^7 - 127/972*a^6 + 17/108*a^5 + 11/36*a^4 + 5/36*a^3 - 5/12*a^2 - 1/3*a - 5/36, 1/2916*a^14 + 1/2916*a^13 - 1/2916*a^12 - 13/1458*a^11 + 1/1458*a^10 + 1/729*a^9 - 20/729*a^8 - 5/81*a^7 - 13/162*a^6 - 13/27*a^5 + 2/9*a^4 + 19/54*a^3 - 37/108*a^2 + 1/4*a + 1/4, 1/8748*a^15 + 1/8748*a^14 - 1/2187*a^13 - 1/4374*a^12 - 49/8748*a^11 + 55/8748*a^10 - 221/8748*a^9 - 73/2916*a^8 - 259/2916*a^7 - 43/324*a^6 - 79/324*a^5 - 61/324*a^4 - 23/162*a^3 - 25/54*a^2 - 53/108*a - 5/12, 1/7479540*a^16 + 83/3739770*a^15 - 101/3739770*a^14 - 112/373977*a^13 + 734/1869885*a^12 + 37489/7479540*a^11 + 64549/7479540*a^10 - 8669/498636*a^9 + 93007/2493180*a^8 - 29729/277020*a^7 + 104137/831060*a^6 - 12727/277020*a^5 - 5047/69255*a^4 - 3799/18468*a^3 + 38807/92340*a^2 - 773/2052*a - 14197/30780, 1/23096108963700*a^17 + 1011097/23096108963700*a^16 - 174325913/11548054481850*a^15 - 240962477/23096108963700*a^14 + 944156173/11548054481850*a^13 - 1177493041/1154805448185*a^12 + 76226785693/23096108963700*a^11 - 53824542227/7698702987900*a^10 + 170559783637/7698702987900*a^9 - 22217510593/2566234329300*a^8 + 422992709/27012992940*a^7 + 126909456157/855411443100*a^6 - 7917594305/17108228862*a^5 - 74027753/142568573850*a^4 + 9236862767/285137147700*a^3 - 5252026529/23761428975*a^2 + 3099029453/95045715900*a + 4111309651/31681905300

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

$C_{3}\times C_{6}$, which has order $18$

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:	$ -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:	$\frac{1296237392}{5774027240925}a^{17}-\frac{5751054466}{5774027240925}a^{16}+\frac{7780720511}{11548054481850}a^{15}+\frac{31761148871}{5774027240925}a^{14}-\frac{50036120147}{23096108963700}a^{13}-\frac{66890394157}{2309610896370}a^{12}+\frac{2219414484809}{23096108963700}a^{11}-\frac{269134188397}{2566234329300}a^{10}-\frac{2804340095659}{7698702987900}a^{9}+\frac{1837704071261}{2566234329300}a^{8}+\frac{388227471937}{513246865860}a^{7}-\frac{805070240413}{285137147700}a^{6}+\frac{496790215169}{171082288620}a^{5}-\frac{26670690797}{31681905300}a^{4}-\frac{408891013469}{285137147700}a^{3}+\frac{52511849647}{95045715900}a^{2}+\frac{1859397227}{47522857950}a-\frac{700792493}{1667468700}$, $\frac{86253005}{307948119516}a^{17}-\frac{219289273}{769870298790}a^{16}-\frac{789681529}{384935149395}a^{15}+\frac{5044647461}{769870298790}a^{14}+\frac{5438934013}{307948119516}a^{13}-\frac{26728426391}{1539740597580}a^{12}+\frac{15943198759}{384935149395}a^{11}+\frac{49029687721}{256623432930}a^{10}-\frac{13275410650}{25662343293}a^{9}-\frac{20801225386}{42770572155}a^{8}+\frac{83378220058}{42770572155}a^{7}-\frac{24389219}{117340390}a^{6}-\frac{180489735973}{57027429540}a^{5}+\frac{48223263653}{9504571590}a^{4}-\frac{1614022267}{1900914318}a^{3}+\frac{126676861}{83373435}a^{2}+\frac{256273747}{1267276212}a+\frac{1526097659}{2112127020}$, $\frac{3489751391}{11548054481850}a^{17}-\frac{276667147}{607792341150}a^{16}-\frac{6302337118}{5774027240925}a^{15}+\frac{60124597883}{11548054481850}a^{14}+\frac{336480729647}{23096108963700}a^{13}-\frac{14653805363}{2309610896370}a^{12}+\frac{1748886110941}{23096108963700}a^{11}+\frac{8832869111}{95045715900}a^{10}-\frac{2876608825541}{7698702987900}a^{9}-\frac{792974829911}{2566234329300}a^{8}+\frac{421471191203}{513246865860}a^{7}-\frac{116829634387}{285137147700}a^{6}+\frac{159576653611}{171082288620}a^{5}+\frac{129700387741}{95045715900}a^{4}+\frac{190069059419}{285137147700}a^{3}+\frac{130611548453}{95045715900}a^{2}-\frac{8640648077}{47522857950}a+\frac{14489983967}{31681905300}$, $\frac{4736462191}{7698702987900}a^{17}-\frac{7865678051}{5774027240925}a^{16}-\frac{46088891783}{23096108963700}a^{15}+\frac{155819126587}{11548054481850}a^{14}+\frac{271118758229}{11548054481850}a^{13}-\frac{105894572647}{2309610896370}a^{12}+\frac{3535366402249}{23096108963700}a^{11}+\frac{3162464023897}{23096108963700}a^{10}-\frac{7640019115699}{7698702987900}a^{9}-\frac{815312198537}{7698702987900}a^{8}+\frac{521557998659}{171082288620}a^{7}-\frac{6131587393937}{2566234329300}a^{6}-\frac{20830356749}{42770572155}a^{5}+\frac{3509865467441}{855411443100}a^{4}-\frac{33362540317}{142568573850}a^{3}+\frac{242810930501}{285137147700}a^{2}+\frac{9273497423}{31681905300}a+\frac{34151237789}{95045715900}$, $\frac{538718167}{11548054481850}a^{17}-\frac{96161171}{11548054481850}a^{16}-\frac{8328168899}{23096108963700}a^{15}+\frac{10154447731}{11548054481850}a^{14}+\frac{86470564639}{23096108963700}a^{13}-\frac{2313566759}{4619221792740}a^{12}+\frac{165800452277}{23096108963700}a^{11}+\frac{336400679287}{7698702987900}a^{10}-\frac{383839870267}{7698702987900}a^{9}-\frac{95446645949}{855411443100}a^{8}+\frac{33039980771}{102649373172}a^{7}+\frac{163280555503}{855411443100}a^{6}-\frac{92780514769}{171082288620}a^{5}+\frac{197324283911}{285137147700}a^{4}+\frac{36154598407}{71284286925}a^{3}-\frac{3746234293}{31681905300}a^{2}+\frac{27713185451}{47522857950}a+\frac{436395779}{880052925}$, $\frac{1889156933}{2566234329300}a^{17}-\frac{841332779}{427705721550}a^{16}-\frac{3267945179}{1283117164650}a^{15}+\frac{12386324221}{641558582325}a^{14}+\frac{63138324743}{2566234329300}a^{13}-\frac{46198017157}{513246865860}a^{12}+\frac{36326560087}{213852860775}a^{11}+\frac{222612755051}{1283117164650}a^{10}-\frac{34146342589}{23761428975}a^{9}+\frac{30544509392}{213852860775}a^{8}+\frac{76267308352}{14256857385}a^{7}-\frac{602292929191}{142568573850}a^{6}-\frac{792862727}{140808468}a^{5}+\frac{370322176003}{47522857950}a^{4}+\frac{1334403492}{293350975}a^{3}-\frac{64685348317}{15840952650}a^{2}-\frac{36748064051}{10560635100}a-\frac{13068153001}{10560635100}$, $\frac{12329263177}{11548054481850}a^{17}-\frac{683577803}{202597447050}a^{16}+\frac{614626241}{23096108963700}a^{15}+\frac{527153436727}{23096108963700}a^{14}+\frac{211825231567}{11548054481850}a^{13}-\frac{105566487853}{1154805448185}a^{12}+\frac{2804457862609}{7698702987900}a^{11}-\frac{2752476733019}{23096108963700}a^{10}-\frac{11747627257127}{7698702987900}a^{9}+\frac{10249049513299}{7698702987900}a^{8}+\frac{1874983043221}{513246865860}a^{7}-\frac{19955942455001}{2566234329300}a^{6}+\frac{5242303649}{704042340}a^{5}-\frac{844697781457}{855411443100}a^{4}+\frac{158147959009}{142568573850}a^{3}+\frac{162414475262}{71284286925}a^{2}+\frac{112337293237}{95045715900}a+\frac{39511157447}{95045715900}$, $\frac{1798138633}{1924675746975}a^{17}-\frac{36798091253}{23096108963700}a^{16}-\frac{98078919761}{23096108963700}a^{15}+\frac{220883773969}{11548054481850}a^{14}+\frac{546333652433}{11548054481850}a^{13}-\frac{253093234637}{4619221792740}a^{12}+\frac{4275528507193}{23096108963700}a^{11}+\frac{7750257064189}{23096108963700}a^{10}-\frac{10848487640473}{7698702987900}a^{9}-\frac{7808127759389}{7698702987900}a^{8}+\frac{825133715099}{171082288620}a^{7}-\frac{2308061590889}{2566234329300}a^{6}-\frac{652231049519}{171082288620}a^{5}+\frac{2072031747121}{427705721550}a^{4}+\frac{307342976158}{71284286925}a^{3}+\frac{473237061887}{285137147700}a^{2}+\frac{74134716521}{31681905300}a+\frac{3403087067}{23761428975}$ 1296237392/5774027240925a^17 - 5751054466/5774027240925a^16 + 7780720511/11548054481850a^15 + 31761148871/5774027240925a^14 - 50036120147/23096108963700a^13 - 66890394157/2309610896370a^12 + 2219414484809/23096108963700a^11 - 269134188397/2566234329300a^10 - 2804340095659/7698702987900a^9 + 1837704071261/2566234329300a^8 + 388227471937/513246865860a^7 - 805070240413/285137147700a^6 + 496790215169/171082288620a^5 - 26670690797/31681905300a^4 - 408891013469/285137147700a^3 + 52511849647/95045715900a^2 + 1859397227/47522857950a - 700792493/1667468700, 86253005/307948119516a^17 - 219289273/769870298790a^16 - 789681529/384935149395a^15 + 5044647461/769870298790a^14 + 5438934013/307948119516a^13 - 26728426391/1539740597580a^12 + 15943198759/384935149395a^11 + 49029687721/256623432930a^10 - 13275410650/25662343293a^9 - 20801225386/42770572155a^8 + 83378220058/42770572155a^7 - 24389219/117340390a^6 - 180489735973/57027429540a^5 + 48223263653/9504571590a^4 - 1614022267/1900914318a^3 + 126676861/83373435a^2 + 256273747/1267276212a + 1526097659/2112127020, 3489751391/11548054481850a^17 - 276667147/607792341150a^16 - 6302337118/5774027240925a^15 + 60124597883/11548054481850a^14 + 336480729647/23096108963700a^13 - 14653805363/2309610896370a^12 + 1748886110941/23096108963700a^11 + 8832869111/95045715900a^10 - 2876608825541/7698702987900a^9 - 792974829911/2566234329300a^8 + 421471191203/513246865860a^7 - 116829634387/285137147700a^6 + 159576653611/171082288620a^5 + 129700387741/95045715900a^4 + 190069059419/285137147700a^3 + 130611548453/95045715900a^2 - 8640648077/47522857950a + 14489983967/31681905300, 4736462191/7698702987900a^17 - 7865678051/5774027240925a^16 - 46088891783/23096108963700a^15 + 155819126587/11548054481850a^14 + 271118758229/11548054481850a^13 - 105894572647/2309610896370a^12 + 3535366402249/23096108963700a^11 + 3162464023897/23096108963700a^10 - 7640019115699/7698702987900a^9 - 815312198537/7698702987900a^8 + 521557998659/171082288620a^7 - 6131587393937/2566234329300a^6 - 20830356749/42770572155a^5 + 3509865467441/855411443100a^4 - 33362540317/142568573850a^3 + 242810930501/285137147700a^2 + 9273497423/31681905300a + 34151237789/95045715900, 538718167/11548054481850a^17 - 96161171/11548054481850a^16 - 8328168899/23096108963700a^15 + 10154447731/11548054481850a^14 + 86470564639/23096108963700a^13 - 2313566759/4619221792740a^12 + 165800452277/23096108963700a^11 + 336400679287/7698702987900a^10 - 383839870267/7698702987900a^9 - 95446645949/855411443100a^8 + 33039980771/102649373172a^7 + 163280555503/855411443100a^6 - 92780514769/171082288620a^5 + 197324283911/285137147700a^4 + 36154598407/71284286925a^3 - 3746234293/31681905300a^2 + 27713185451/47522857950a + 436395779/880052925, 1889156933/2566234329300a^17 - 841332779/427705721550a^16 - 3267945179/1283117164650a^15 + 12386324221/641558582325a^14 + 63138324743/2566234329300a^13 - 46198017157/513246865860a^12 + 36326560087/213852860775a^11 + 222612755051/1283117164650a^10 - 34146342589/23761428975a^9 + 30544509392/213852860775a^8 + 76267308352/14256857385a^7 - 602292929191/142568573850a^6 - 792862727/140808468a^5 + 370322176003/47522857950a^4 + 1334403492/293350975a^3 - 64685348317/15840952650a^2 - 36748064051/10560635100a - 13068153001/10560635100, 12329263177/11548054481850a^17 - 683577803/202597447050a^16 + 614626241/23096108963700a^15 + 527153436727/23096108963700a^14 + 211825231567/11548054481850a^13 - 105566487853/1154805448185a^12 + 2804457862609/7698702987900a^11 - 2752476733019/23096108963700a^10 - 11747627257127/7698702987900a^9 + 10249049513299/7698702987900a^8 + 1874983043221/513246865860a^7 - 19955942455001/2566234329300a^6 + 5242303649/704042340a^5 - 844697781457/855411443100a^4 + 158147959009/142568573850a^3 + 162414475262/71284286925a^2 + 112337293237/95045715900a + 39511157447/95045715900, 1798138633/1924675746975a^17 - 36798091253/23096108963700a^16 - 98078919761/23096108963700a^15 + 220883773969/11548054481850a^14 + 546333652433/11548054481850a^13 - 253093234637/4619221792740a^12 + 4275528507193/23096108963700a^11 + 7750257064189/23096108963700a^10 - 10848487640473/7698702987900a^9 - 7808127759389/7698702987900a^8 + 825133715099/171082288620a^7 - 2308061590889/2566234329300a^6 - 652231049519/171082288620a^5 + 2072031747121/427705721550a^4 + 307342976158/71284286925a^3 + 473237061887/285137147700a^2 + 74134716521/31681905300a + 3403087067/23761428975	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:	$ 1281109.48365 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1281109.48365 \cdot 18}{2\cdot\sqrt{242190571378050467501912064}}\cr\approx \mathstrut & 11.3075614071 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729)

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 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729, 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 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729);

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 - 4*x^16 + 22*x^15 + 44*x^14 - 74*x^13 + 223*x^12 + 312*x^11 - 1620*x^10 - 576*x^9 + 5508*x^8 - 2916*x^7 - 3861*x^6 + 7614*x^5 + 2916*x^4 - 486*x^3 + 2916*x^2 + 1458*x + 729);

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{-339}) $, 3.1.1356.3 x3, 3.1.339.1 x3, 3.1.1356.2 x3, 3.1.1356.1 x3, 6.0.623331504.1, 6.0.38958219.1, 6.0.623331504.3, 6.0.623331504.2, 9.1.845237519424.1 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:	9.1.845237519424.1
Minimal sibling:	9.1.845237519424.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.3.0.1}{3} }^{6}$	${\href{/padicField/7.3.0.1}{3} }^{6}$	${\href{/padicField/11.2.0.1}{2} }^{9}$	${\href{/padicField/13.3.0.1}{3} }^{6}$	${\href{/padicField/17.3.0.1}{3} }^{6}$	${\href{/padicField/19.2.0.1}{2} }^{9}$	${\href{/padicField/23.3.0.1}{3} }^{6}$	${\href{/padicField/29.3.0.1}{3} }^{6}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.2.0.1}{2} }^{9}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.2.0.1}{2} }^{9}$	${\href{/padicField/47.3.0.1}{3} }^{6}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.3.0.1}{3} }^{6}$

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	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$113$ 113	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.2	$x^{2} + 339$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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