LMFDB - Number field 18.0.6347285018761982937208599123.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27)

gp: K = bnfinit(y^18 - 9*y^17 + 39*y^16 - 108*y^15 + 225*y^14 - 399*y^13 + 832*y^12 - 2067*y^11 + 4146*y^10 - 5715*y^9 + 4383*y^8 + 417*y^7 + 535*y^6 - 10974*y^5 + 19080*y^4 - 15975*y^3 + 6264*y^2 - 675*y + 27, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27)

$ x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 225 x^{14} - 399 x^{13} + 832 x^{12} - 2067 x^{11} + \cdots + 27 $ x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27

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:	$-6347285018761982937208599123$ -6347285018761982937208599123 $\medspace = -\,3^{9}\cdot 7^{12}\cdot 13^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$35.04$	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:	$3^{1/2}7^{2/3}13^{2/3}\approx 35.04194650073235$
Ramified primes:	$3$, $7$, $13$ 3, 7, 13	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{5}$, $\frac{1}{27}a^{12}-\frac{1}{27}a^{10}-\frac{1}{9}a^{7}-\frac{4}{27}a^{6}-\frac{4}{9}a^{5}+\frac{7}{27}a^{4}-\frac{4}{9}a^{3}+\frac{2}{9}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{11}-\frac{1}{9}a^{8}-\frac{4}{27}a^{7}-\frac{1}{9}a^{6}+\frac{7}{27}a^{5}-\frac{1}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{297}a^{14}+\frac{4}{297}a^{13}+\frac{5}{297}a^{12}-\frac{16}{297}a^{11}-\frac{4}{99}a^{10}-\frac{2}{99}a^{9}+\frac{38}{297}a^{8}-\frac{28}{297}a^{7}-\frac{2}{297}a^{6}+\frac{37}{297}a^{5}+\frac{41}{99}a^{4}+\frac{5}{11}a^{3}-\frac{13}{33}a^{2}-\frac{7}{33}a-\frac{2}{11}$, $\frac{1}{297}a^{15}-\frac{1}{99}a^{12}+\frac{8}{297}a^{11}+\frac{1}{33}a^{10}-\frac{4}{297}a^{9}-\frac{5}{99}a^{8}-\frac{1}{9}a^{7}-\frac{7}{99}a^{6}-\frac{113}{297}a^{5}-\frac{20}{99}a^{4}-\frac{43}{99}a^{3}-\frac{10}{33}a^{2}-\frac{1}{3}a-\frac{3}{11}$, $\frac{1}{560136951}a^{16}-\frac{8}{560136951}a^{15}-\frac{745190}{560136951}a^{14}+\frac{5216470}{560136951}a^{13}-\frac{8214512}{560136951}a^{12}-\frac{18527402}{560136951}a^{11}-\frac{24687832}{560136951}a^{10}-\frac{18061573}{560136951}a^{9}+\frac{8671346}{560136951}a^{8}+\frac{30175490}{560136951}a^{7}-\frac{50141101}{560136951}a^{6}+\frac{172485506}{560136951}a^{5}-\frac{10051280}{186712317}a^{4}-\frac{3691384}{16973847}a^{3}-\frac{697990}{62237439}a^{2}+\frac{627073}{5657949}a+\frac{1205579}{6915271}$, $\frac{1}{208931082723}a^{17}+\frac{178}{208931082723}a^{16}-\frac{36580355}{208931082723}a^{15}-\frac{216372122}{208931082723}a^{14}+\frac{339674518}{208931082723}a^{13}-\frac{2808149261}{208931082723}a^{12}-\frac{8238549628}{208931082723}a^{11}+\frac{751851344}{208931082723}a^{10}+\frac{8182004813}{208931082723}a^{9}-\frac{10557378181}{208931082723}a^{8}-\frac{4271015323}{208931082723}a^{7}-\frac{26699059129}{208931082723}a^{6}-\frac{9999077293}{23214564747}a^{5}-\frac{7529772746}{23214564747}a^{4}-\frac{5517563509}{23214564747}a^{3}-\frac{2484790973}{7738188249}a^{2}+\frac{1364842108}{7738188249}a-\frac{2442302173}{7738188249}$ 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, 1/3*a^8 - 1/3*a^2, 1/9*a^9 - 1/3*a^5 - 4/9*a^3 - 1/3*a, 1/9*a^10 - 1/9*a^4, 1/9*a^11 - 1/9*a^5, 1/27*a^12 - 1/27*a^10 - 1/9*a^7 - 4/27*a^6 - 4/9*a^5 + 7/27*a^4 - 4/9*a^3 + 2/9*a^2 - 1/3*a + 1/3, 1/27*a^13 - 1/27*a^11 - 1/9*a^8 - 4/27*a^7 - 1/9*a^6 + 7/27*a^5 - 1/9*a^4 + 2/9*a^3 + 1/3*a, 1/297*a^14 + 4/297*a^13 + 5/297*a^12 - 16/297*a^11 - 4/99*a^10 - 2/99*a^9 + 38/297*a^8 - 28/297*a^7 - 2/297*a^6 + 37/297*a^5 + 41/99*a^4 + 5/11*a^3 - 13/33*a^2 - 7/33*a - 2/11, 1/297*a^15 - 1/99*a^12 + 8/297*a^11 + 1/33*a^10 - 4/297*a^9 - 5/99*a^8 - 1/9*a^7 - 7/99*a^6 - 113/297*a^5 - 20/99*a^4 - 43/99*a^3 - 10/33*a^2 - 1/3*a - 3/11, 1/560136951*a^16 - 8/560136951*a^15 - 745190/560136951*a^14 + 5216470/560136951*a^13 - 8214512/560136951*a^12 - 18527402/560136951*a^11 - 24687832/560136951*a^10 - 18061573/560136951*a^9 + 8671346/560136951*a^8 + 30175490/560136951*a^7 - 50141101/560136951*a^6 + 172485506/560136951*a^5 - 10051280/186712317*a^4 - 3691384/16973847*a^3 - 697990/62237439*a^2 + 627073/5657949*a + 1205579/6915271, 1/208931082723*a^17 + 178/208931082723*a^16 - 36580355/208931082723*a^15 - 216372122/208931082723*a^14 + 339674518/208931082723*a^13 - 2808149261/208931082723*a^12 - 8238549628/208931082723*a^11 + 751851344/208931082723*a^10 + 8182004813/208931082723*a^9 - 10557378181/208931082723*a^8 - 4271015323/208931082723*a^7 - 26699059129/208931082723*a^6 - 9999077293/23214564747*a^5 - 7529772746/23214564747*a^4 - 5517563509/23214564747*a^3 - 2484790973/7738188249*a^2 + 1364842108/7738188249*a - 2442302173/7738188249

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

$C_{6}\times C_{6}$, which has order $36$ (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{273282}{21317323} a^{17} - \frac{2322897}{21317323} a^{16} + \frac{257657692}{575567721} a^{15} - \frac{226022770}{191855907} a^{14} + \frac{1360044094}{575567721} a^{13} - \frac{787051954}{191855907} a^{12} + \frac{5145629038}{575567721} a^{11} - \frac{4331616916}{191855907} a^{10} + \frac{24822274466}{575567721} a^{9} - \frac{10540719332}{191855907} a^{8} + \frac{19882022810}{575567721} a^{7} + \frac{3046293190}{191855907} a^{6} + \frac{10170609104}{575567721} a^{5} - \frac{24505195901}{191855907} a^{4} + \frac{35562483478}{191855907} a^{3} - \frac{2792178854}{21317323} a^{2} + \frac{2405792693}{63951969} a - \frac{35650823}{21317323} $ (273282)/(21317323)a^(17) - (2322897)/(21317323)a^(16) + (257657692)/(575567721)a^(15) - (226022770)/(191855907)a^(14) + (1360044094)/(575567721)a^(13) - (787051954)/(191855907)a^(12) + (5145629038)/(575567721)a^(11) - (4331616916)/(191855907)a^(10) + (24822274466)/(575567721)a^(9) - (10540719332)/(191855907)a^(8) + (19882022810)/(575567721)a^(7) + (3046293190)/(191855907)a^(6) + (10170609104)/(575567721)a^(5) - (24505195901)/(191855907)a^(4) + (35562483478)/(191855907)a^(3) - (2792178854)/(21317323)a^(2) + (2405792693)/(63951969)*a - (35650823)/(21317323) (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:	$\frac{683502451}{208931082723}a^{17}-\frac{5440436864}{208931082723}a^{16}+\frac{20682949186}{208931082723}a^{15}-\frac{49670473373}{208931082723}a^{14}+\frac{8327850101}{18993734793}a^{13}-\frac{13711004041}{18993734793}a^{12}+\frac{360359108369}{208931082723}a^{11}-\frac{950433370696}{208931082723}a^{10}+\frac{1648661872391}{208931082723}a^{9}-\frac{1681559822512}{208931082723}a^{8}+\frac{303797348813}{208931082723}a^{7}+\frac{1692732896513}{208931082723}a^{6}+\frac{536252785648}{69643694241}a^{5}-\frac{79393842166}{2579396083}a^{4}+\frac{644928832909}{23214564747}a^{3}-\frac{18441379950}{2579396083}a^{2}-\frac{35600497687}{7738188249}a+\frac{2693633861}{7738188249}$, $\frac{1232152114}{208931082723}a^{17}-\frac{10593695131}{208931082723}a^{16}+\frac{44111878136}{208931082723}a^{15}-\frac{118065158272}{208931082723}a^{14}+\frac{240864741578}{208931082723}a^{13}-\frac{423203012203}{208931082723}a^{12}+\frac{33692549090}{7738188249}a^{11}-\frac{2272881192248}{208931082723}a^{10}+\frac{4398514594054}{208931082723}a^{9}-\frac{5809434477380}{208931082723}a^{8}+\frac{4071753737998}{208931082723}a^{7}+\frac{987203937829}{208931082723}a^{6}+\frac{1554307137644}{208931082723}a^{5}-\frac{1347558751862}{23214564747}a^{4}+\frac{6364137643111}{69643694241}a^{3}-\frac{580679116103}{7738188249}a^{2}+\frac{612324284846}{23214564747}a-\frac{10657593134}{7738188249}$, $\frac{809569646}{208931082723}a^{17}-\frac{6099769354}{208931082723}a^{16}+\frac{22259832338}{208931082723}a^{15}-\frac{52045256302}{208931082723}a^{14}+\frac{96046741724}{208931082723}a^{13}-\frac{159901294876}{208931082723}a^{12}+\frac{398388115630}{208931082723}a^{11}-\frac{1021177548728}{208931082723}a^{10}+\frac{1686233360689}{208931082723}a^{9}-\frac{1713738890738}{208931082723}a^{8}+\frac{335397419248}{208931082723}a^{7}+\frac{1503422057851}{208931082723}a^{6}+\frac{284602366154}{23214564747}a^{5}-\frac{1945869192950}{69643694241}a^{4}+\frac{642092363156}{23214564747}a^{3}-\frac{236926532815}{23214564747}a^{2}-\frac{3407161631}{2579396083}a-\frac{491954926}{7738188249}$, $\frac{501862334}{208931082723}a^{17}-\frac{4308734537}{208931082723}a^{16}+\frac{18085324462}{208931082723}a^{15}-\frac{49087525394}{208931082723}a^{14}+\frac{101608108480}{208931082723}a^{13}-\frac{179781489056}{208931082723}a^{12}+\frac{4729980450}{2579396083}a^{11}-\frac{945485023468}{208931082723}a^{10}+\frac{1849831803059}{208931082723}a^{9}-\frac{2527157017600}{208931082723}a^{8}+\frac{1890276674804}{208931082723}a^{7}+\frac{225445838573}{208931082723}a^{6}+\frac{592419999190}{208931082723}a^{5}-\frac{1572287545115}{69643694241}a^{4}+\frac{2882229617504}{69643694241}a^{3}-\frac{849634157641}{23214564747}a^{2}+\frac{274393381624}{23214564747}a-\frac{1939653445}{2579396083}$, $\frac{35999309}{18993734793}a^{17}-\frac{101286296}{6331244931}a^{16}+\frac{410525699}{6331244931}a^{15}-\frac{353613461}{2110414977}a^{14}+\frac{31222151}{94496193}a^{13}-\frac{3593635688}{6331244931}a^{12}+\frac{23730785456}{18993734793}a^{11}-\frac{6734284402}{2110414977}a^{10}+\frac{12695954444}{2110414977}a^{9}-\frac{15543603487}{2110414977}a^{8}+\frac{25872100228}{6331244931}a^{7}+\frac{1861617724}{575567721}a^{6}+\frac{42837380273}{18993734793}a^{5}-\frac{120733872007}{6331244931}a^{4}+\frac{163922075002}{6331244931}a^{3}-\frac{33462969635}{2110414977}a^{2}+\frac{3940273976}{2110414977}a+\frac{969267809}{703471659}$, $\frac{58830026}{18993734793}a^{17}-\frac{192722936}{7738188249}a^{16}+\frac{6754630151}{69643694241}a^{15}-\frac{16889579959}{69643694241}a^{14}+\frac{10913398690}{23214564747}a^{13}-\frac{56055251080}{69643694241}a^{12}+\frac{387281835475}{208931082723}a^{11}-\frac{328567743529}{69643694241}a^{10}+\frac{590508316688}{69643694241}a^{9}-\frac{689813449571}{69643694241}a^{8}+\frac{112250902138}{23214564747}a^{7}+\frac{323460675217}{69643694241}a^{6}+\frac{1541083295683}{208931082723}a^{5}-\frac{1849407256649}{69643694241}a^{4}+\frac{190390396285}{6331244931}a^{3}-\frac{369155633266}{23214564747}a^{2}+\frac{2596768337}{2110414977}a-\frac{376580981}{7738188249}$, $\frac{338836109}{69643694241}a^{17}-\frac{2487914771}{69643694241}a^{16}+\frac{2893503643}{23214564747}a^{15}-\frac{19094094230}{69643694241}a^{14}+\frac{33382335935}{69643694241}a^{13}-\frac{2016103608}{2579396083}a^{12}+\frac{48868562719}{23214564747}a^{11}-\frac{42690423703}{7738188249}a^{10}+\frac{193032252953}{23214564747}a^{9}-\frac{474377509468}{69643694241}a^{8}-\frac{98284601111}{69643694241}a^{7}+\frac{225860836886}{23214564747}a^{6}+\frac{1363161944383}{69643694241}a^{5}-\frac{219415750856}{6331244931}a^{4}+\frac{331231784495}{23214564747}a^{3}-\frac{7177522075}{2110414977}a^{2}+\frac{4627485838}{7738188249}a+\frac{22656400}{703471659}$, $\frac{338836109}{69643694241}a^{17}-\frac{3272299082}{69643694241}a^{16}+\frac{553910571}{2579396083}a^{15}-\frac{42972351305}{69643694241}a^{14}+\frac{90716331920}{69643694241}a^{13}-\frac{53439475592}{23214564747}a^{12}+\frac{107360468572}{23214564747}a^{11}-\frac{267983120525}{23214564747}a^{10}+\frac{187547512996}{7738188249}a^{9}-\frac{2392817456323}{69643694241}a^{8}+\frac{1856223143974}{69643694241}a^{7}+\frac{79661003452}{23214564747}a^{6}-\frac{402910969880}{69643694241}a^{5}-\frac{4518034952758}{69643694241}a^{4}+\frac{2884743398588}{23214564747}a^{3}-\frac{2088440547662}{23214564747}a^{2}+\frac{209445380812}{7738188249}a-\frac{18123341827}{7738188249}$ 683502451/208931082723a^17 - 5440436864/208931082723a^16 + 20682949186/208931082723a^15 - 49670473373/208931082723a^14 + 8327850101/18993734793a^13 - 13711004041/18993734793a^12 + 360359108369/208931082723a^11 - 950433370696/208931082723a^10 + 1648661872391/208931082723a^9 - 1681559822512/208931082723a^8 + 303797348813/208931082723a^7 + 1692732896513/208931082723a^6 + 536252785648/69643694241a^5 - 79393842166/2579396083a^4 + 644928832909/23214564747a^3 - 18441379950/2579396083a^2 - 35600497687/7738188249a + 2693633861/7738188249, 1232152114/208931082723a^17 - 10593695131/208931082723a^16 + 44111878136/208931082723a^15 - 118065158272/208931082723a^14 + 240864741578/208931082723a^13 - 423203012203/208931082723a^12 + 33692549090/7738188249a^11 - 2272881192248/208931082723a^10 + 4398514594054/208931082723a^9 - 5809434477380/208931082723a^8 + 4071753737998/208931082723a^7 + 987203937829/208931082723a^6 + 1554307137644/208931082723a^5 - 1347558751862/23214564747a^4 + 6364137643111/69643694241a^3 - 580679116103/7738188249a^2 + 612324284846/23214564747a - 10657593134/7738188249, 809569646/208931082723a^17 - 6099769354/208931082723a^16 + 22259832338/208931082723a^15 - 52045256302/208931082723a^14 + 96046741724/208931082723a^13 - 159901294876/208931082723a^12 + 398388115630/208931082723a^11 - 1021177548728/208931082723a^10 + 1686233360689/208931082723a^9 - 1713738890738/208931082723a^8 + 335397419248/208931082723a^7 + 1503422057851/208931082723a^6 + 284602366154/23214564747a^5 - 1945869192950/69643694241a^4 + 642092363156/23214564747a^3 - 236926532815/23214564747a^2 - 3407161631/2579396083a - 491954926/7738188249, 501862334/208931082723a^17 - 4308734537/208931082723a^16 + 18085324462/208931082723a^15 - 49087525394/208931082723a^14 + 101608108480/208931082723a^13 - 179781489056/208931082723a^12 + 4729980450/2579396083a^11 - 945485023468/208931082723a^10 + 1849831803059/208931082723a^9 - 2527157017600/208931082723a^8 + 1890276674804/208931082723a^7 + 225445838573/208931082723a^6 + 592419999190/208931082723a^5 - 1572287545115/69643694241a^4 + 2882229617504/69643694241a^3 - 849634157641/23214564747a^2 + 274393381624/23214564747a - 1939653445/2579396083, 35999309/18993734793a^17 - 101286296/6331244931a^16 + 410525699/6331244931a^15 - 353613461/2110414977a^14 + 31222151/94496193a^13 - 3593635688/6331244931a^12 + 23730785456/18993734793a^11 - 6734284402/2110414977a^10 + 12695954444/2110414977a^9 - 15543603487/2110414977a^8 + 25872100228/6331244931a^7 + 1861617724/575567721a^6 + 42837380273/18993734793a^5 - 120733872007/6331244931a^4 + 163922075002/6331244931a^3 - 33462969635/2110414977a^2 + 3940273976/2110414977a + 969267809/703471659, 58830026/18993734793a^17 - 192722936/7738188249a^16 + 6754630151/69643694241a^15 - 16889579959/69643694241a^14 + 10913398690/23214564747a^13 - 56055251080/69643694241a^12 + 387281835475/208931082723a^11 - 328567743529/69643694241a^10 + 590508316688/69643694241a^9 - 689813449571/69643694241a^8 + 112250902138/23214564747a^7 + 323460675217/69643694241a^6 + 1541083295683/208931082723a^5 - 1849407256649/69643694241a^4 + 190390396285/6331244931a^3 - 369155633266/23214564747a^2 + 2596768337/2110414977a - 376580981/7738188249, 338836109/69643694241a^17 - 2487914771/69643694241a^16 + 2893503643/23214564747a^15 - 19094094230/69643694241a^14 + 33382335935/69643694241a^13 - 2016103608/2579396083a^12 + 48868562719/23214564747a^11 - 42690423703/7738188249a^10 + 193032252953/23214564747a^9 - 474377509468/69643694241a^8 - 98284601111/69643694241a^7 + 225860836886/23214564747a^6 + 1363161944383/69643694241a^5 - 219415750856/6331244931a^4 + 331231784495/23214564747a^3 - 7177522075/2110414977a^2 + 4627485838/7738188249a + 22656400/703471659, 338836109/69643694241a^17 - 3272299082/69643694241a^16 + 553910571/2579396083a^15 - 42972351305/69643694241a^14 + 90716331920/69643694241a^13 - 53439475592/23214564747a^12 + 107360468572/23214564747a^11 - 267983120525/23214564747a^10 + 187547512996/7738188249a^9 - 2392817456323/69643694241a^8 + 1856223143974/69643694241a^7 + 79661003452/23214564747a^6 - 402910969880/69643694241a^5 - 4518034952758/69643694241a^4 + 2884743398588/23214564747a^3 - 2088440547662/23214564747a^2 + 209445380812/7738188249a - 18123341827/7738188249 (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:	$ 5151205.65339 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 5151205.65339 \cdot 36}{6\cdot\sqrt{6347285018761982937208599123}}\cr\approx \mathstrut & 5.92086220780 \end{aligned}\] (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 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27)

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 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27, 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 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27);

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 + 39*x^16 - 108*x^15 + 225*x^14 - 399*x^13 + 832*x^12 - 2067*x^11 + 4146*x^10 - 5715*x^9 + 4383*x^8 + 417*x^7 + 535*x^6 - 10974*x^5 + 19080*x^4 - 15975*x^3 + 6264*x^2 - 675*x + 27);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_3\times S_3$ (as 18T3):

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 9 conjugacy class representatives for $S_3 \times C_3$

Character table for $S_3 \times C_3$

Intermediate fields

$\Q(\sqrt{-3}) $, 3.1.24843.1 x3, 3.3.8281.2, 6.0.1851523947.3, 6.0.1851523947.1, 6.0.223587.1 x2, 9.3.15332469805107.3 x3

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 6 sibling:	6.0.223587.1
Degree 9 sibling:	9.3.15332469805107.3
Minimal sibling:	6.0.223587.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{3}$	R	${\href{/padicField/5.6.0.1}{6} }^{3}$	R	${\href{/padicField/11.2.0.1}{2} }^{9}$	R	${\href{/padicField/17.6.0.1}{6} }^{3}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.6.0.1}{6} }^{3}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.6.0.1}{6} }^{3}$	${\href{/padicField/43.3.0.1}{3} }^{6}$	${\href{/padicField/47.6.0.1}{6} }^{3}$	${\href{/padicField/53.6.0.1}{6} }^{3}$	${\href{/padicField/59.6.0.1}{6} }^{3}$

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

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

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

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

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

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$13$ 13	13.9.6.1	$x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$13$ 13	13.9.6.1	$x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$

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