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Properties

Label 18.4.613...408.1
Degree $18$
Signature $[4, 7]$
Discriminant $-6.135\times 10^{28}$
Root discriminant \(39.75\)
Ramified primes $2,3,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^4:(C_6^2:C_4)$ (as 18T576)

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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 - 18*x^16 - 12*x^15 + 117*x^14 + 180*x^13 - 210*x^12 - 912*x^11 - 1071*x^10 + 1200*x^9 + 5742*x^8 + 5748*x^7 - 921*x^6 - 3468*x^5 + 306*x^4 + 984*x^3 - 342*x^2 + 24*x + 4)
 
Copy content gp:K = bnfinit(y^18 - 18*y^16 - 12*y^15 + 117*y^14 + 180*y^13 - 210*y^12 - 912*y^11 - 1071*y^10 + 1200*y^9 + 5742*y^8 + 5748*y^7 - 921*y^6 - 3468*y^5 + 306*y^4 + 984*y^3 - 342*y^2 + 24*y + 4, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 18*x^16 - 12*x^15 + 117*x^14 + 180*x^13 - 210*x^12 - 912*x^11 - 1071*x^10 + 1200*x^9 + 5742*x^8 + 5748*x^7 - 921*x^6 - 3468*x^5 + 306*x^4 + 984*x^3 - 342*x^2 + 24*x + 4);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 18*x^16 - 12*x^15 + 117*x^14 + 180*x^13 - 210*x^12 - 912*x^11 - 1071*x^10 + 1200*x^9 + 5742*x^8 + 5748*x^7 - 921*x^6 - 3468*x^5 + 306*x^4 + 984*x^3 - 342*x^2 + 24*x + 4)
 

\( x^{18} - 18 x^{16} - 12 x^{15} + 117 x^{14} + 180 x^{13} - 210 x^{12} - 912 x^{11} - 1071 x^{10} + \cdots + 4 \) 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:  $[4, 7]$
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:   \(-61351691829216883959983505408\) \(\medspace = -\,2^{46}\cdot 3^{26}\cdot 7^{3}\) 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:  \(39.75\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{32}a^{14}-\frac{1}{8}a^{13}-\frac{3}{16}a^{10}-\frac{1}{4}a^{9}+\frac{1}{16}a^{8}-\frac{9}{32}a^{6}-\frac{3}{8}a^{5}-\frac{5}{16}a^{4}-\frac{1}{2}a^{3}+\frac{3}{16}a^{2}-\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{32}a^{15}-\frac{3}{16}a^{11}+\frac{1}{16}a^{9}-\frac{1}{4}a^{8}-\frac{9}{32}a^{7}-\frac{1}{2}a^{6}-\frac{5}{16}a^{5}-\frac{1}{4}a^{4}+\frac{3}{16}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{32}a^{16}+\frac{1}{16}a^{12}-\frac{3}{16}a^{10}-\frac{1}{4}a^{9}-\frac{1}{32}a^{8}-\frac{1}{2}a^{7}-\frac{1}{16}a^{6}-\frac{1}{4}a^{5}+\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{80\cdots 52}a^{17}-\frac{66738842292777}{80\cdots 52}a^{16}+\frac{60759559313585}{40\cdots 76}a^{15}+\frac{46397977253827}{80\cdots 52}a^{14}+\frac{1016052567329}{9476737860688}a^{13}+\frac{169088504138707}{40\cdots 76}a^{12}+\frac{322800149155215}{40\cdots 76}a^{11}+\frac{244770118525165}{20\cdots 88}a^{10}+\frac{825170884448515}{80\cdots 52}a^{9}+\frac{734860527666151}{80\cdots 52}a^{8}+\frac{996676738467899}{20\cdots 88}a^{7}-\frac{27\cdots 05}{80\cdots 52}a^{6}+\frac{13\cdots 43}{40\cdots 76}a^{5}+\frac{4565717762279}{33168582512408}a^{4}-\frac{272126010511307}{10\cdots 44}a^{3}+\frac{3875211662971}{15745397145968}a^{2}+\frac{46614016106107}{505820883314222}a+\frac{110446810378025}{20\cdots 88}$ 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:  Trivial group, which has order $1$ (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:  Trivial group, which has order $1$ (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:  $10$
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:   \( -1 \)  (order $2$) 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:   $\frac{123946119}{1559876872}a^{17}+\frac{161327991}{3119753744}a^{16}-\frac{4518036753}{3119753744}a^{15}-\frac{2859884823}{1559876872}a^{14}+\frac{114105465}{12785876}a^{13}+\frac{30835614999}{1559876872}a^{12}-\frac{13911582663}{1559876872}a^{11}-\frac{127182102951}{1559876872}a^{10}-\frac{24701950738}{194984609}a^{9}+\frac{136495803717}{3119753744}a^{8}+\frac{1591242629661}{3119753744}a^{7}+\frac{279234539301}{389969218}a^{6}+\frac{314375735475}{1559876872}a^{5}-\frac{6530238765}{25571752}a^{4}-\frac{134325029895}{1559876872}a^{3}+\frac{11869754874}{194984609}a^{2}-\frac{5578385895}{779938436}a-\frac{183057709}{389969218}$, $\frac{106786262236753}{11\cdots 36}a^{17}-\frac{9928731614657}{72260126187746}a^{16}-\frac{16\cdots 95}{11\cdots 36}a^{15}+\frac{12\cdots 59}{11\cdots 36}a^{14}+\frac{12582828691113}{1353819694384}a^{13}+\frac{207158137687577}{72260126187746}a^{12}-\frac{69\cdots 25}{289040504750984}a^{11}-\frac{28\cdots 85}{578081009501968}a^{10}-\frac{29\cdots 11}{11\cdots 36}a^{9}+\frac{87\cdots 43}{578081009501968}a^{8}+\frac{35\cdots 85}{11\cdots 36}a^{7}+\frac{78\cdots 05}{11\cdots 36}a^{6}-\frac{15\cdots 21}{72260126187746}a^{5}-\frac{422905444904939}{9476737860688}a^{4}+\frac{41\cdots 09}{578081009501968}a^{3}-\frac{49600533436659}{2249342449424}a^{2}+\frac{614862808362433}{289040504750984}a+\frac{1258238333361}{289040504750984}$, $\frac{5435918070703}{165166002714848}a^{17}-\frac{14294972850311}{82583001357424}a^{16}-\frac{53647705407357}{165166002714848}a^{15}+\frac{47340541283689}{20645750339356}a^{14}+\frac{2433655415075}{1353819694384}a^{13}-\frac{451634805578037}{41291500678712}a^{12}-\frac{504748652047171}{41291500678712}a^{11}+\frac{543786372540077}{41291500678712}a^{10}+\frac{92\cdots 39}{165166002714848}a^{9}+\frac{72\cdots 95}{82583001357424}a^{8}-\frac{14\cdots 25}{165166002714848}a^{7}-\frac{15\cdots 19}{41291500678712}a^{6}-\frac{72\cdots 69}{41291500678712}a^{5}+\frac{155008753874925}{676909847192}a^{4}+\frac{70\cdots 27}{82583001357424}a^{3}-\frac{7485620760449}{80333658908}a^{2}+\frac{706660534007617}{41291500678712}a-\frac{1763875649301}{10322875169678}$, $\frac{11\cdots 95}{80\cdots 52}a^{17}+\frac{528364836606177}{80\cdots 52}a^{16}-\frac{640853056223460}{252910441657111}a^{15}-\frac{22\cdots 23}{80\cdots 52}a^{14}+\frac{151993469245399}{9476737860688}a^{13}+\frac{12\cdots 93}{40\cdots 76}a^{12}-\frac{85\cdots 33}{40\cdots 76}a^{11}-\frac{28\cdots 53}{20\cdots 88}a^{10}-\frac{16\cdots 75}{80\cdots 52}a^{9}+\frac{93\cdots 25}{80\cdots 52}a^{8}+\frac{35\cdots 01}{40\cdots 76}a^{7}+\frac{89\cdots 37}{80\cdots 52}a^{6}+\frac{51\cdots 99}{40\cdots 76}a^{5}-\frac{18\cdots 37}{33168582512408}a^{4}-\frac{25\cdots 59}{20\cdots 88}a^{3}+\frac{22\cdots 33}{15745397145968}a^{2}-\frac{12\cdots 07}{10\cdots 44}a-\frac{34\cdots 37}{20\cdots 88}$, $\frac{46446885454419}{578081009501968}a^{17}+\frac{5567047563859}{144520252375492}a^{16}-\frac{837801151104383}{578081009501968}a^{15}-\frac{18\cdots 51}{11\cdots 36}a^{14}+\frac{1516508193567}{169227461798}a^{13}+\frac{674832426546752}{36130063093873}a^{12}-\frac{14\cdots 31}{144520252375492}a^{11}-\frac{46\cdots 35}{578081009501968}a^{10}-\frac{69\cdots 65}{578081009501968}a^{9}+\frac{30\cdots 69}{578081009501968}a^{8}+\frac{28\cdots 37}{578081009501968}a^{7}+\frac{77\cdots 99}{11\cdots 36}a^{6}+\frac{47\cdots 41}{289040504750984}a^{5}-\frac{24\cdots 93}{9476737860688}a^{4}-\frac{24\cdots 19}{289040504750984}a^{3}+\frac{138387295557223}{2249342449424}a^{2}-\frac{115775497533989}{72260126187746}a-\frac{115643006869529}{289040504750984}$, $\frac{580793121842529}{20\cdots 88}a^{17}-\frac{586809085443557}{20\cdots 88}a^{16}-\frac{11\cdots 97}{252910441657111}a^{15}+\frac{11\cdots 73}{10\cdots 44}a^{14}+\frac{35991082847091}{1184592232586}a^{13}+\frac{57\cdots 36}{252910441657111}a^{12}-\frac{17\cdots 65}{252910441657111}a^{11}-\frac{47\cdots 42}{252910441657111}a^{10}-\frac{30\cdots 47}{20\cdots 88}a^{9}+\frac{87\cdots 79}{20\cdots 88}a^{8}+\frac{29\cdots 07}{252910441657111}a^{7}+\frac{68\cdots 41}{10\cdots 44}a^{6}-\frac{53\cdots 23}{10\cdots 44}a^{5}-\frac{67\cdots 57}{16584291256204}a^{4}+\frac{51\cdots 11}{252910441657111}a^{3}+\frac{93952670800449}{1968174643246}a^{2}-\frac{91\cdots 71}{252910441657111}a+\frac{13\cdots 53}{252910441657111}$, $\frac{997248445911455}{80\cdots 52}a^{17}-\frac{588223109311955}{80\cdots 52}a^{16}-\frac{87\cdots 77}{40\cdots 76}a^{15}-\frac{895368432248023}{40\cdots 76}a^{14}+\frac{138201569374805}{9476737860688}a^{13}+\frac{56\cdots 09}{40\cdots 76}a^{12}-\frac{14\cdots 27}{40\cdots 76}a^{11}-\frac{37\cdots 89}{40\cdots 76}a^{10}-\frac{59\cdots 27}{80\cdots 52}a^{9}+\frac{16\cdots 83}{80\cdots 52}a^{8}+\frac{11\cdots 31}{20\cdots 88}a^{7}+\frac{69\cdots 53}{20\cdots 88}a^{6}-\frac{14\cdots 01}{40\cdots 76}a^{5}-\frac{15\cdots 31}{66337165024816}a^{4}+\frac{31\cdots 27}{10\cdots 44}a^{3}+\frac{446104094480963}{3936349286492}a^{2}-\frac{94\cdots 55}{10\cdots 44}a+\frac{18\cdots 11}{10\cdots 44}$, $\frac{16093687891075}{11\cdots 36}a^{17}-\frac{4335433996231}{11\cdots 36}a^{16}-\frac{266392358960589}{11\cdots 36}a^{15}-\frac{146792919790099}{11\cdots 36}a^{14}+\frac{1831312325551}{1353819694384}a^{13}+\frac{13\cdots 85}{578081009501968}a^{12}-\frac{467641061434147}{289040504750984}a^{11}-\frac{31\cdots 17}{289040504750984}a^{10}-\frac{18\cdots 65}{11\cdots 36}a^{9}+\frac{11\cdots 41}{11\cdots 36}a^{8}+\frac{78\cdots 27}{11\cdots 36}a^{7}+\frac{10\cdots 77}{11\cdots 36}a^{6}+\frac{12\cdots 45}{36130063093873}a^{5}-\frac{75000319779405}{4738368930344}a^{4}-\frac{60\cdots 25}{578081009501968}a^{3}+\frac{7130750119261}{2249342449424}a^{2}+\frac{374041115356059}{289040504750984}a-\frac{105326574950925}{289040504750984}$, $\frac{18904570978691}{578081009501968}a^{17}+\frac{129369140721059}{11\cdots 36}a^{16}-\frac{928180402847729}{11\cdots 36}a^{15}-\frac{24\cdots 71}{11\cdots 36}a^{14}+\frac{1989475282037}{338454923596}a^{13}+\frac{96\cdots 51}{578081009501968}a^{12}-\frac{47\cdots 23}{578081009501968}a^{11}-\frac{87\cdots 07}{144520252375492}a^{10}-\frac{11\cdots 77}{144520252375492}a^{9}+\frac{40\cdots 11}{11\cdots 36}a^{8}+\frac{44\cdots 49}{11\cdots 36}a^{7}+\frac{55\cdots 97}{11\cdots 36}a^{6}-\frac{65\cdots 47}{578081009501968}a^{5}-\frac{479181343798865}{1184592232586}a^{4}+\frac{42\cdots 01}{578081009501968}a^{3}+\frac{348649857334237}{2249342449424}a^{2}-\frac{22\cdots 81}{289040504750984}a+\frac{32\cdots 95}{289040504750984}$, $\frac{366043639952299}{40\cdots 76}a^{17}+\frac{79610575009293}{80\cdots 52}a^{16}-\frac{13\cdots 23}{80\cdots 52}a^{15}-\frac{49\cdots 07}{40\cdots 76}a^{14}+\frac{52703875534977}{4738368930344}a^{13}+\frac{69\cdots 49}{40\cdots 76}a^{12}-\frac{86\cdots 41}{40\cdots 76}a^{11}-\frac{35\cdots 09}{40\cdots 76}a^{10}-\frac{19\cdots 93}{20\cdots 88}a^{9}+\frac{99\cdots 11}{80\cdots 52}a^{8}+\frac{45\cdots 67}{80\cdots 52}a^{7}+\frac{10\cdots 47}{20\cdots 88}a^{6}-\frac{73\cdots 59}{40\cdots 76}a^{5}-\frac{29\cdots 31}{66337165024816}a^{4}-\frac{40\cdots 33}{40\cdots 76}a^{3}+\frac{387796824361407}{3936349286492}a^{2}-\frac{58\cdots 37}{20\cdots 88}a+\frac{16\cdots 25}{10\cdots 44}$ 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:  \( 537112098.848 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{7}\cdot 537112098.848 \cdot 1}{2\cdot\sqrt{61351691829216883959983505408}}\cr\approx \mathstrut & 6.70657412674 \end{aligned}\] (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 - 18*x^16 - 12*x^15 + 117*x^14 + 180*x^13 - 210*x^12 - 912*x^11 - 1071*x^10 + 1200*x^9 + 5742*x^8 + 5748*x^7 - 921*x^6 - 3468*x^5 + 306*x^4 + 984*x^3 - 342*x^2 + 24*x + 4) 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 - 18*x^16 - 12*x^15 + 117*x^14 + 180*x^13 - 210*x^12 - 912*x^11 - 1071*x^10 + 1200*x^9 + 5742*x^8 + 5748*x^7 - 921*x^6 - 3468*x^5 + 306*x^4 + 984*x^3 - 342*x^2 + 24*x + 4, 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 - 18*x^16 - 12*x^15 + 117*x^14 + 180*x^13 - 210*x^12 - 912*x^11 - 1071*x^10 + 1200*x^9 + 5742*x^8 + 5748*x^7 - 921*x^6 - 3468*x^5 + 306*x^4 + 984*x^3 - 342*x^2 + 24*x + 4); 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 - 18*x^16 - 12*x^15 + 117*x^14 + 180*x^13 - 210*x^12 - 912*x^11 - 1071*x^10 + 1200*x^9 + 5742*x^8 + 5748*x^7 - 921*x^6 - 3468*x^5 + 306*x^4 + 984*x^3 - 342*x^2 + 24*x + 4); 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

$C_3^4:(C_6^2:C_4)$ (as 18T576):

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 11664
The 49 conjugacy class representatives for $C_3^4:(C_6^2:C_4)$
Character table for $C_3^4:(C_6^2:C_4)$

Intermediate fields

\(\Q(\sqrt{2}) \), 6.2.11943936.2

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 18 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ R ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.1.4.11a1.12$x^{4} + 8 x^{3} + 4 x^{2} + 18$$4$$1$$11$$C_4$$$[3, 4]$$
2.1.4.8b1.6$x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$$[2, 3]$$
2.1.8.24c1.61$x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$$[2, 3, 4]$$
\(3\) Copy content Toggle raw display 3.2.9.26b30.1$x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4176 x^{14} + 14112 x^{13} + 38304 x^{12} + 85251 x^{11} + 157569 x^{10} + 243578 x^{9} + 315756 x^{8} + 342795 x^{7} + 309978 x^{6} + 231078 x^{5} + 139584 x^{4} + 66396 x^{3} + 23640 x^{2} + 5688 x + 707$$9$$2$$26$not computednot computed
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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