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Properties

Label 18.0.327...375.4
Degree $18$
Signature $[0, 9]$
Discriminant $-3.274\times 10^{32}$
Root discriminant \(64.03\)
Ramified primes $3,5,7$
Class number $27$ (GRH)
Class group [3, 9] (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 + 15*x^16 - 70*x^15 + 225*x^14 - 2100*x^13 + 5475*x^12 - 31500*x^11 + 145125*x^10 - 285250*x^9 + 2491875*x^8 - 2940000*x^7 + 17585625*x^6 - 86231250*x^5 + 165112500*x^4 - 199675000*x^3 + 330750000*x^2 - 514500000*x + 343000000)
 
Copy content gp:K = bnfinit(y^18 + 15*y^16 - 70*y^15 + 225*y^14 - 2100*y^13 + 5475*y^12 - 31500*y^11 + 145125*y^10 - 285250*y^9 + 2491875*y^8 - 2940000*y^7 + 17585625*y^6 - 86231250*y^5 + 165112500*y^4 - 199675000*y^3 + 330750000*y^2 - 514500000*y + 343000000, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 15*x^16 - 70*x^15 + 225*x^14 - 2100*x^13 + 5475*x^12 - 31500*x^11 + 145125*x^10 - 285250*x^9 + 2491875*x^8 - 2940000*x^7 + 17585625*x^6 - 86231250*x^5 + 165112500*x^4 - 199675000*x^3 + 330750000*x^2 - 514500000*x + 343000000);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 15*x^16 - 70*x^15 + 225*x^14 - 2100*x^13 + 5475*x^12 - 31500*x^11 + 145125*x^10 - 285250*x^9 + 2491875*x^8 - 2940000*x^7 + 17585625*x^6 - 86231250*x^5 + 165112500*x^4 - 199675000*x^3 + 330750000*x^2 - 514500000*x + 343000000)
 

\( x^{18} + 15 x^{16} - 70 x^{15} + 225 x^{14} - 2100 x^{13} + 5475 x^{12} - 31500 x^{11} + \cdots + 343000000 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $18$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 9]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-327356346842831533869771240234375\) \(\medspace = -\,3^{24}\cdot 5^{12}\cdot 7^{15}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(64.03\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{4/3}5^{2/3}7^{5/6}\approx 64.03096433292906$
Ramified primes:   \(3\), \(5\), \(7\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_3\times S_3$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\zeta_{7})\)

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

$1$, $a$, $a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{25}a^{6}$, $\frac{1}{50}a^{7}-\frac{1}{50}a^{6}-\frac{1}{10}a^{5}-\frac{1}{10}a^{4}-\frac{1}{10}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{100}a^{8}-\frac{1}{100}a^{7}+\frac{1}{100}a^{6}+\frac{1}{20}a^{5}-\frac{1}{20}a^{4}-\frac{1}{20}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{500}a^{9}-\frac{1}{50}a^{6}-\frac{1}{10}a^{4}-\frac{1}{10}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{500}a^{10}-\frac{1}{50}a^{6}+\frac{1}{20}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{500}a^{11}-\frac{1}{50}a^{6}-\frac{1}{10}a^{5}-\frac{1}{20}a^{4}-\frac{1}{2}a$, $\frac{1}{2500}a^{12}+\frac{1}{20}a^{5}-\frac{1}{10}a^{4}-\frac{1}{10}a^{3}-\frac{1}{2}a$, $\frac{1}{2500}a^{13}+\frac{1}{100}a^{6}-\frac{1}{10}a^{5}-\frac{1}{10}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{17500}a^{14}+\frac{1}{17500}a^{12}+\frac{3}{3500}a^{10}+\frac{1}{350}a^{8}-\frac{1}{100}a^{7}-\frac{3}{175}a^{6}+\frac{1}{20}a^{5}+\frac{3}{70}a^{4}+\frac{1}{20}a^{3}-\frac{5}{14}a^{2}-\frac{1}{2}a$, $\frac{1}{1184788325000}a^{15}+\frac{150169}{5923941625}a^{14}+\frac{2534359}{47391533000}a^{13}+\frac{4704027}{23695766500}a^{12}+\frac{44785631}{47391533000}a^{11}-\frac{753027}{947830660}a^{10}+\frac{5474797}{9478306600}a^{9}-\frac{3702047}{1184788325}a^{8}-\frac{19138183}{9478306600}a^{7}+\frac{28694929}{4739153300}a^{6}-\frac{34626601}{379132264}a^{5}+\frac{17216739}{189566132}a^{4}-\frac{150185269}{1895661320}a^{3}+\frac{4691663}{94783066}a^{2}+\frac{3364924}{6770219}a-\frac{906083}{6770219}$, $\frac{1}{33\cdots 00}a^{16}+\frac{61989}{23\cdots 00}a^{15}-\frac{137990922507953}{66\cdots 00}a^{14}+\frac{5176487649041}{67\cdots 00}a^{13}+\frac{373031957755369}{66\cdots 00}a^{12}+\frac{9438111767219}{47\cdots 00}a^{11}+\frac{13\cdots 27}{13\cdots 00}a^{10}-\frac{1256802627614}{23\cdots 75}a^{9}+\frac{3770123962173}{815025935772496}a^{8}-\frac{9395128026381}{27\cdots 00}a^{7}+\frac{17987572543667}{26\cdots 00}a^{6}-\frac{634528857591}{189202449375758}a^{5}+\frac{46\cdots 81}{52\cdots 40}a^{4}-\frac{12215453537817}{540578426787880}a^{3}+\frac{4350176430095}{54057842678788}a^{2}+\frac{1799061950641}{13514460669697}a-\frac{3726876576936}{13514460669697}$, $\frac{1}{66\cdots 00}a^{17}-\frac{25595653}{66\cdots 00}a^{15}-\frac{24694340247557}{94\cdots 00}a^{14}-\frac{599160720491687}{13\cdots 00}a^{13}-\frac{971375264779}{94\cdots 00}a^{12}+\frac{24\cdots 11}{26\cdots 00}a^{11}+\frac{40915528380647}{94\cdots 00}a^{10}+\frac{15\cdots 01}{26\cdots 00}a^{9}-\frac{144616995430651}{37\cdots 00}a^{8}-\frac{44\cdots 17}{52\cdots 00}a^{7}-\frac{6625901974784}{337861516742425}a^{6}-\frac{95\cdots 59}{10\cdots 80}a^{5}-\frac{563493940191419}{75\cdots 20}a^{4}-\frac{1501192767}{133035050890}a^{3}-\frac{17319662067185}{54057842678788}a^{2}-\frac{2545066306688}{13514460669697}a+\frac{757035304357}{13514460669697}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  $C_{3}\times C_{9}$, which has order $27$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{3}\times C_{9}$, which has order $27$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $8$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{556726623741}{33110428640757650000} a^{17} + \frac{169373875893}{8277607160189412500} a^{16} + \frac{9126153725333}{33110428640757650000} a^{15} - \frac{2832238114767}{3311042864075765000} a^{14} + \frac{17941120101159}{6622085728151530000} a^{13} - \frac{26633520910013}{827760716018941250} a^{12} + \frac{71045472587049}{1324417145630306000} a^{11} - \frac{153905939262891}{331104286407576500} a^{10} + \frac{2513087454028559}{1324417145630306000} a^{9} - \frac{332586988936569}{132441714563030600} a^{8} + \frac{797127193602321}{20375648394312400} a^{7} - \frac{222974844241791}{66220857281515300} a^{6} + \frac{15453082092627603}{52976685825212240} a^{5} - \frac{29842023760511769}{26488342912606120} a^{4} + \frac{5295646972201137}{3784048987515160} a^{3} - \frac{167966664470154}{94601224687879} a^{2} + \frac{54986402946558}{13514460669697} a - \frac{54970537621502}{13514460669697} \)  (order $14$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{466389795411}{66\cdots 00}a^{17}+\frac{437489204979}{33\cdots 00}a^{16}+\frac{8274975166417}{66\cdots 00}a^{15}-\frac{434000665701}{16\cdots 00}a^{14}+\frac{12641575741791}{13\cdots 00}a^{13}-\frac{83875897456131}{66\cdots 00}a^{12}+\frac{6688573645977}{52\cdots 00}a^{11}-\frac{18643538119509}{10\cdots 00}a^{10}+\frac{16\cdots 91}{26\cdots 00}a^{9}-\frac{34868676220821}{66\cdots 00}a^{8}+\frac{330202833098157}{21\cdots 96}a^{7}+\frac{30\cdots 73}{26\cdots 00}a^{6}+\frac{26\cdots 63}{21\cdots 96}a^{5}-\frac{23\cdots 83}{66\cdots 30}a^{4}+\frac{10\cdots 91}{37\cdots 60}a^{3}-\frac{152775441448839}{378404898751516}a^{2}+\frac{7552160415747}{27028921339394}a-\frac{613108347097}{13514460669697}$, $\frac{451858817187}{33\cdots 00}a^{17}+\frac{443992029069}{16\cdots 00}a^{16}+\frac{8205266069449}{33\cdots 00}a^{15}-\frac{32368117221}{63\cdots 50}a^{14}+\frac{12336817597359}{66\cdots 00}a^{13}-\frac{82761748866451}{33\cdots 00}a^{12}+\frac{33115037600493}{13\cdots 00}a^{11}-\frac{241825113477729}{66\cdots 00}a^{10}+\frac{16\cdots 87}{13\cdots 00}a^{9}-\frac{37743822544437}{33\cdots 50}a^{8}+\frac{82\cdots 69}{26\cdots 00}a^{7}+\frac{27\cdots 97}{13\cdots 00}a^{6}+\frac{13\cdots 51}{52\cdots 40}a^{5}-\frac{23\cdots 23}{33\cdots 65}a^{4}+\frac{11\cdots 39}{18\cdots 80}a^{3}-\frac{68598883755801}{54057842678788}a^{2}+\frac{63173750227317}{27028921339394}a-\frac{15112156439859}{13514460669697}$, $\frac{242220199521}{33\cdots 00}a^{17}-\frac{49074753841}{20\cdots 25}a^{16}+\frac{1739626576841}{33\cdots 00}a^{15}-\frac{2770684988473}{33\cdots 00}a^{14}+\frac{1233894803487}{50\cdots 00}a^{13}-\frac{27120836349493}{16\cdots 00}a^{12}+\frac{20057411779889}{26\cdots 00}a^{11}-\frac{7841757082857}{33\cdots 50}a^{10}+\frac{19\cdots 63}{13\cdots 00}a^{9}-\frac{486566724483601}{13\cdots 00}a^{8}+\frac{43\cdots 77}{26\cdots 00}a^{7}-\frac{41\cdots 13}{66\cdots 00}a^{6}+\frac{30\cdots 39}{52\cdots 40}a^{5}-\frac{22\cdots 53}{26\cdots 20}a^{4}+\frac{88\cdots 27}{37\cdots 60}a^{3}-\frac{19449496777579}{27028921339394}a^{2}-\frac{58791473140723}{27028921339394}a+\frac{15552938730128}{13514460669697}$, $\frac{191352780559}{66\cdots 00}a^{17}-\frac{276711300941}{33\cdots 00}a^{16}+\frac{217356130977}{26\cdots 00}a^{15}-\frac{39254766157}{12\cdots 00}a^{14}+\frac{4967947512667}{26\cdots 00}a^{13}-\frac{66175831501683}{66\cdots 00}a^{12}+\frac{106350303845477}{26\cdots 00}a^{11}-\frac{290259834811201}{13\cdots 00}a^{10}+\frac{22\cdots 51}{26\cdots 00}a^{9}-\frac{203956575907263}{66\cdots 00}a^{8}+\frac{15\cdots 61}{10\cdots 80}a^{7}-\frac{19\cdots 31}{52\cdots 40}a^{6}+\frac{17\cdots 39}{10\cdots 80}a^{5}-\frac{282394591472135}{662208572815153}a^{4}+\frac{824644996424707}{473006123439395}a^{3}-\frac{921589965602707}{189202449375758}a^{2}+\frac{94150964536319}{13514460669697}a-\frac{57228025408446}{13514460669697}$, $\frac{105259054007}{66\cdots 00}a^{17}+\frac{1148697009923}{41\cdots 50}a^{16}+\frac{11087667400931}{41\cdots 50}a^{15}-\frac{2865489871862}{41\cdots 25}a^{14}+\frac{8353051587576}{41\cdots 25}a^{13}-\frac{242367107236909}{82\cdots 50}a^{12}+\frac{56117902413059}{16\cdots 50}a^{11}-\frac{13\cdots 53}{33\cdots 00}a^{10}+\frac{401794113030743}{25\cdots 00}a^{9}-\frac{217829704385066}{16\cdots 25}a^{8}+\frac{23\cdots 11}{66\cdots 30}a^{7}+\frac{53\cdots 23}{33\cdots 50}a^{6}+\frac{87\cdots 69}{33\cdots 65}a^{5}-\frac{63\cdots 87}{66\cdots 30}a^{4}+\frac{23\cdots 41}{37\cdots 60}a^{3}-\frac{45\cdots 83}{378404898751516}a^{2}+\frac{831637295041079}{27028921339394}a-\frac{15811607657377}{1039573897669}$, $\frac{681852870893}{82\cdots 00}a^{17}+\frac{7136645693721}{33\cdots 00}a^{16}+\frac{1290673585667}{82\cdots 50}a^{15}-\frac{14719783678781}{66\cdots 00}a^{14}+\frac{26805000235659}{33\cdots 00}a^{13}-\frac{38005153959127}{26\cdots 00}a^{12}+\frac{14975188762621}{33\cdots 00}a^{11}-\frac{536699396103953}{26\cdots 00}a^{10}+\frac{20\cdots 67}{33\cdots 00}a^{9}-\frac{6608136331835}{10\cdots 48}a^{8}+\frac{24\cdots 49}{13\cdots 00}a^{7}+\frac{65\cdots 99}{26\cdots 00}a^{6}+\frac{10\cdots 51}{66\cdots 30}a^{5}-\frac{19\cdots 71}{52\cdots 40}a^{4}-\frac{270396846360223}{756809797503032}a^{3}-\frac{524202547589900}{94601224687879}a^{2}+\frac{144478530412482}{13514460669697}a+\frac{36783472084848}{13514460669697}$, $\frac{25556485706097}{66\cdots 00}a^{17}-\frac{18365050285191}{12\cdots 00}a^{16}+\frac{33\cdots 63}{33\cdots 00}a^{15}-\frac{213276945593027}{82\cdots 50}a^{14}+\frac{78\cdots 01}{66\cdots 00}a^{13}-\frac{24\cdots 37}{33\cdots 00}a^{12}+\frac{22\cdots 23}{13\cdots 00}a^{11}-\frac{65\cdots 67}{66\cdots 00}a^{10}+\frac{34\cdots 37}{13\cdots 00}a^{9}-\frac{24\cdots 54}{16\cdots 25}a^{8}+\frac{87\cdots 87}{26\cdots 00}a^{7}+\frac{69\cdots 91}{13\cdots 00}a^{6}-\frac{15\cdots 31}{52\cdots 40}a^{5}+\frac{16\cdots 27}{33\cdots 65}a^{4}-\frac{10\cdots 91}{145540345673660}a^{3}+\frac{26\cdots 63}{189202449375758}a^{2}-\frac{49\cdots 89}{27028921339394}a+\frac{12\cdots 44}{13514460669697}$, $\frac{51162521514207}{16\cdots 00}a^{17}-\frac{121173612699351}{82\cdots 00}a^{16}+\frac{12\cdots 89}{16\cdots 00}a^{15}-\frac{70258376998213}{41\cdots 25}a^{14}+\frac{22\cdots 71}{33\cdots 00}a^{13}-\frac{78\cdots 01}{16\cdots 00}a^{12}+\frac{47\cdots 89}{66\cdots 00}a^{11}-\frac{15\cdots 07}{33\cdots 00}a^{10}+\frac{59\cdots 31}{66\cdots 00}a^{9}+\frac{35\cdots 67}{33\cdots 50}a^{8}-\frac{20\cdots 59}{52\cdots 24}a^{7}+\frac{67\cdots 13}{66\cdots 00}a^{6}-\frac{30\cdots 01}{52\cdots 24}a^{5}+\frac{38\cdots 62}{34134462516245}a^{4}-\frac{12\cdots 69}{946012246878790}a^{3}+\frac{21\cdots 29}{94601224687879}a^{2}-\frac{97\cdots 65}{27028921339394}a+\frac{31\cdots 64}{13514460669697}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 294941269.205 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 294941269.205 \cdot 27}{14\cdot\sqrt{327356346842831533869771240234375}}\cr\approx \mathstrut & 0.479821501834 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 + 15*x^16 - 70*x^15 + 225*x^14 - 2100*x^13 + 5475*x^12 - 31500*x^11 + 145125*x^10 - 285250*x^9 + 2491875*x^8 - 2940000*x^7 + 17585625*x^6 - 86231250*x^5 + 165112500*x^4 - 199675000*x^3 + 330750000*x^2 - 514500000*x + 343000000) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 + 15*x^16 - 70*x^15 + 225*x^14 - 2100*x^13 + 5475*x^12 - 31500*x^11 + 145125*x^10 - 285250*x^9 + 2491875*x^8 - 2940000*x^7 + 17585625*x^6 - 86231250*x^5 + 165112500*x^4 - 199675000*x^3 + 330750000*x^2 - 514500000*x + 343000000, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 15*x^16 - 70*x^15 + 225*x^14 - 2100*x^13 + 5475*x^12 - 31500*x^11 + 145125*x^10 - 285250*x^9 + 2491875*x^8 - 2940000*x^7 + 17585625*x^6 - 86231250*x^5 + 165112500*x^4 - 199675000*x^3 + 330750000*x^2 - 514500000*x + 343000000); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 15*x^16 - 70*x^15 + 225*x^14 - 2100*x^13 + 5475*x^12 - 31500*x^11 + 145125*x^10 - 285250*x^9 + 2491875*x^8 - 2940000*x^7 + 17585625*x^6 - 86231250*x^5 + 165112500*x^4 - 199675000*x^3 + 330750000*x^2 - 514500000*x + 343000000); 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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content 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

\(\Q(\sqrt{-7}) \), 3.1.14175.1 x3, \(\Q(\zeta_{7})^+\), 6.0.1406514375.2, \(\Q(\zeta_{7})\), 6.0.68919204375.2 x2, 9.3.6838508054109375.21 x3

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 6 sibling: 6.0.68919204375.2
Degree 9 sibling: 9.3.6838508054109375.21
Minimal sibling: 6.0.68919204375.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{6}$ R R R ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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

Copy content comment:Frobenius cycle types
 
Copy content 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)]
 
Copy content 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]])
 
Copy content 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))];
 
Copy content 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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.6.3.24a1.1$x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 29 x^{12} + 24 x^{11} + 51 x^{10} + 36 x^{9} + 72 x^{8} + 60 x^{7} + 85 x^{6} + 78 x^{5} + 69 x^{4} + 44 x^{3} + 60 x^{2} + 48 x + 23$$3$$6$$24$$S_3 \times C_3$not computed
\(5\) Copy content Toggle raw display 5.6.3.12a1.2$x^{18} + 3 x^{16} + 12 x^{15} + 6 x^{14} + 24 x^{13} + 61 x^{12} + 36 x^{11} + 66 x^{10} + 136 x^{9} + 69 x^{8} + 60 x^{7} + 121 x^{6} + 48 x^{5} + 18 x^{4} + 48 x^{3} + 12 x^{2} + 13$$3$$6$$12$$S_3 \times C_3$$$[\ ]_{3}^{6}$$
\(7\) Copy content Toggle raw display 7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$
7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$
7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)