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Properties

Label 18.4.146...375.1
Degree $18$
Signature $(4, 7)$
Discriminant $-1.462\times 10^{37}$
Root discriminant \(116.07\)
Ramified primes $5,7,41,419,449$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^6.C_2\wr C_6$ (as 18T858)

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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 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125)
 
Copy content gp:K = bnfinit(y^18 - 9*y^17 - 23*y^16 + 300*y^15 + 515*y^14 - 4561*y^13 - 10821*y^12 + 31931*y^11 + 128094*y^10 - 8849*y^9 - 656070*y^8 - 1052332*y^7 + 297296*y^6 + 3208702*y^5 + 5247365*y^4 + 4745832*y^3 + 2696725*y^2 + 963650*y + 201125, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125)
 

\( x^{18} - 9 x^{17} - 23 x^{16} + 300 x^{15} + 515 x^{14} - 4561 x^{13} - 10821 x^{12} + 31931 x^{11} + \cdots + 201125 \) 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:   \(-14616700658977641856215848806693687375\) \(\medspace = -\,5^{3}\cdot 7^{12}\cdot 41^{4}\cdot 419^{3}\cdot 449^{4}\) 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:  \(116.07\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}7^{2/3}41^{2/3}419^{1/2}449^{2/3}\approx 116773.46634277423$
Ramified primes:   \(5\), \(7\), \(41\), \(419\), \(449\) 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:  \(\Q(\sqrt{-2095}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{105}a^{16}-\frac{4}{105}a^{15}+\frac{22}{105}a^{14}+\frac{3}{7}a^{13}-\frac{1}{3}a^{12}-\frac{26}{105}a^{11}+\frac{4}{105}a^{10}+\frac{1}{5}a^{9}+\frac{44}{105}a^{8}-\frac{19}{105}a^{7}+\frac{1}{21}a^{6}+\frac{4}{15}a^{5}-\frac{19}{105}a^{4}+\frac{52}{105}a^{3}-\frac{8}{21}a^{2}-\frac{11}{35}a-\frac{5}{21}$, $\frac{1}{12\cdots 75}a^{17}+\frac{52\cdots 26}{12\cdots 75}a^{16}-\frac{47\cdots 84}{18\cdots 25}a^{15}-\frac{93\cdots 46}{25\cdots 95}a^{14}+\frac{57\cdots 28}{25\cdots 95}a^{13}-\frac{10\cdots 04}{14\cdots 75}a^{12}-\frac{19\cdots 77}{42\cdots 25}a^{11}-\frac{53\cdots 54}{12\cdots 75}a^{10}-\frac{26\cdots 71}{12\cdots 75}a^{9}+\frac{35\cdots 22}{42\cdots 25}a^{8}+\frac{13\cdots 72}{28\cdots 55}a^{7}-\frac{54\cdots 82}{12\cdots 75}a^{6}+\frac{35\cdots 76}{12\cdots 75}a^{5}-\frac{57\cdots 34}{18\cdots 25}a^{4}+\frac{13\cdots 07}{25\cdots 95}a^{3}-\frac{49\cdots 68}{12\cdots 75}a^{2}-\frac{89\cdots 58}{36\cdots 85}a-\frac{14\cdots 03}{50\cdots 19}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

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:   $\frac{59\cdots 24}{76\cdots 25}a^{17}-\frac{65\cdots 76}{76\cdots 25}a^{16}-\frac{32\cdots 12}{76\cdots 25}a^{15}+\frac{41\cdots 51}{15\cdots 65}a^{14}-\frac{13\cdots 23}{15\cdots 65}a^{13}-\frac{11\cdots 88}{25\cdots 75}a^{12}-\frac{12\cdots 48}{25\cdots 75}a^{11}+\frac{31\cdots 04}{76\cdots 25}a^{10}+\frac{27\cdots 21}{76\cdots 25}a^{9}-\frac{17\cdots 74}{85\cdots 25}a^{8}-\frac{27\cdots 63}{73\cdots 65}a^{7}+\frac{23\cdots 32}{76\cdots 25}a^{6}+\frac{10\cdots 74}{76\cdots 25}a^{5}+\frac{67\cdots 38}{76\cdots 25}a^{4}-\frac{11\cdots 67}{15\cdots 65}a^{3}-\frac{10\cdots 32}{76\cdots 25}a^{2}-\frac{11\cdots 69}{15\cdots 65}a-\frac{56\cdots 70}{30\cdots 53}$, $\frac{59\cdots 24}{76\cdots 25}a^{17}-\frac{65\cdots 76}{76\cdots 25}a^{16}-\frac{32\cdots 12}{76\cdots 25}a^{15}+\frac{41\cdots 51}{15\cdots 65}a^{14}-\frac{13\cdots 23}{15\cdots 65}a^{13}-\frac{11\cdots 88}{25\cdots 75}a^{12}-\frac{12\cdots 48}{25\cdots 75}a^{11}+\frac{31\cdots 04}{76\cdots 25}a^{10}+\frac{27\cdots 21}{76\cdots 25}a^{9}-\frac{17\cdots 74}{85\cdots 25}a^{8}-\frac{27\cdots 63}{73\cdots 65}a^{7}+\frac{23\cdots 32}{76\cdots 25}a^{6}+\frac{10\cdots 74}{76\cdots 25}a^{5}+\frac{67\cdots 38}{76\cdots 25}a^{4}-\frac{11\cdots 67}{15\cdots 65}a^{3}-\frac{10\cdots 32}{76\cdots 25}a^{2}-\frac{11\cdots 69}{15\cdots 65}a-\frac{25\cdots 17}{30\cdots 53}$, $\frac{39\cdots 24}{12\cdots 75}a^{17}-\frac{47\cdots 76}{12\cdots 75}a^{16}+\frac{36\cdots 88}{12\cdots 75}a^{15}+\frac{23\cdots 41}{25\cdots 95}a^{14}-\frac{29\cdots 68}{25\cdots 95}a^{13}-\frac{55\cdots 38}{42\cdots 25}a^{12}+\frac{88\cdots 84}{14\cdots 75}a^{11}+\frac{14\cdots 79}{12\cdots 75}a^{10}+\frac{85\cdots 21}{12\cdots 75}a^{9}-\frac{20\cdots 47}{42\cdots 25}a^{8}-\frac{10\cdots 48}{12\cdots 95}a^{7}+\frac{39\cdots 57}{12\cdots 75}a^{6}+\frac{31\cdots 74}{12\cdots 75}a^{5}+\frac{38\cdots 13}{12\cdots 75}a^{4}+\frac{41\cdots 13}{25\cdots 95}a^{3}+\frac{24\cdots 43}{12\cdots 75}a^{2}-\frac{27\cdots 34}{25\cdots 95}a+\frac{27\cdots 50}{50\cdots 19}$, $\frac{19\cdots 67}{14\cdots 75}a^{17}-\frac{22\cdots 28}{14\cdots 75}a^{16}+\frac{20\cdots 09}{14\cdots 75}a^{15}+\frac{21\cdots 33}{56\cdots 91}a^{14}-\frac{12\cdots 14}{28\cdots 55}a^{13}-\frac{70\cdots 87}{14\cdots 75}a^{12}+\frac{17\cdots 68}{14\cdots 75}a^{11}+\frac{57\cdots 02}{14\cdots 75}a^{10}+\frac{60\cdots 48}{14\cdots 75}a^{9}-\frac{19\cdots 08}{14\cdots 75}a^{8}-\frac{16\cdots 44}{40\cdots 65}a^{7}-\frac{28\cdots 19}{14\cdots 75}a^{6}+\frac{11\cdots 32}{14\cdots 75}a^{5}+\frac{25\cdots 09}{14\cdots 75}a^{4}+\frac{53\cdots 91}{28\cdots 55}a^{3}+\frac{16\cdots 69}{14\cdots 75}a^{2}+\frac{25\cdots 69}{56\cdots 91}a+\frac{48\cdots 37}{56\cdots 91}$, $\frac{26\cdots 99}{25\cdots 95}a^{17}-\frac{52\cdots 94}{50\cdots 19}a^{16}-\frac{42\cdots 06}{25\cdots 95}a^{15}+\frac{86\cdots 27}{25\cdots 95}a^{14}+\frac{13\cdots 40}{50\cdots 19}a^{13}-\frac{45\cdots 73}{84\cdots 65}a^{12}-\frac{40\cdots 30}{56\cdots 91}a^{11}+\frac{11\cdots 93}{25\cdots 95}a^{10}+\frac{26\cdots 57}{25\cdots 95}a^{9}-\frac{16\cdots 12}{12\cdots 95}a^{8}-\frac{55\cdots 58}{84\cdots 65}a^{7}-\frac{10\cdots 33}{25\cdots 95}a^{6}+\frac{30\cdots 07}{25\cdots 95}a^{5}+\frac{65\cdots 09}{25\cdots 95}a^{4}+\frac{57\cdots 92}{25\cdots 95}a^{3}+\frac{41\cdots 59}{36\cdots 85}a^{2}+\frac{92\cdots 57}{25\cdots 95}a+\frac{16\cdots 92}{50\cdots 19}$, $\frac{15\cdots 46}{12\cdots 75}a^{17}-\frac{15\cdots 39}{12\cdots 75}a^{16}-\frac{21\cdots 83}{12\cdots 75}a^{15}+\frac{19\cdots 41}{50\cdots 19}a^{14}+\frac{90\cdots 59}{36\cdots 85}a^{13}-\frac{85\cdots 09}{14\cdots 75}a^{12}-\frac{30\cdots 47}{42\cdots 25}a^{11}+\frac{92\cdots 93}{18\cdots 25}a^{10}+\frac{14\cdots 49}{12\cdots 75}a^{9}-\frac{70\cdots 68}{42\cdots 25}a^{8}-\frac{21\cdots 21}{28\cdots 55}a^{7}-\frac{77\cdots 21}{18\cdots 25}a^{6}+\frac{19\cdots 16}{12\cdots 75}a^{5}+\frac{40\cdots 17}{12\cdots 75}a^{4}+\frac{55\cdots 28}{25\cdots 95}a^{3}-\frac{20\cdots 28}{12\cdots 75}a^{2}-\frac{55\cdots 52}{50\cdots 19}a-\frac{37\cdots 32}{72\cdots 17}$, $\frac{53\cdots 62}{60\cdots 75}a^{17}-\frac{44\cdots 91}{42\cdots 25}a^{16}+\frac{39\cdots 83}{42\cdots 25}a^{15}+\frac{70\cdots 97}{28\cdots 55}a^{14}-\frac{24\cdots 48}{84\cdots 65}a^{13}-\frac{20\cdots 82}{60\cdots 75}a^{12}+\frac{41\cdots 71}{42\cdots 25}a^{11}+\frac{40\cdots 63}{14\cdots 75}a^{10}+\frac{15\cdots 48}{60\cdots 75}a^{9}-\frac{47\cdots 31}{42\cdots 25}a^{8}-\frac{22\cdots 63}{84\cdots 65}a^{7}-\frac{18\cdots 38}{42\cdots 25}a^{6}+\frac{34\cdots 12}{60\cdots 75}a^{5}+\frac{44\cdots 33}{42\cdots 25}a^{4}+\frac{84\cdots 28}{84\cdots 65}a^{3}+\frac{80\cdots 71}{14\cdots 75}a^{2}+\frac{17\cdots 16}{84\cdots 65}a+\frac{23\cdots 26}{56\cdots 91}$, $\frac{27\cdots 29}{12\cdots 75}a^{17}-\frac{34\cdots 46}{12\cdots 75}a^{16}+\frac{38\cdots 48}{12\cdots 75}a^{15}+\frac{16\cdots 71}{25\cdots 95}a^{14}-\frac{31\cdots 18}{25\cdots 95}a^{13}-\frac{37\cdots 73}{42\cdots 25}a^{12}+\frac{48\cdots 42}{42\cdots 25}a^{11}+\frac{95\cdots 09}{12\cdots 75}a^{10}-\frac{27\cdots 34}{12\cdots 75}a^{9}-\frac{51\cdots 79}{14\cdots 75}a^{8}-\frac{32\cdots 73}{12\cdots 95}a^{7}+\frac{84\cdots 72}{12\cdots 75}a^{6}+\frac{16\cdots 54}{12\cdots 75}a^{5}+\frac{88\cdots 23}{12\cdots 75}a^{4}-\frac{72\cdots 57}{25\cdots 95}a^{3}-\frac{80\cdots 22}{12\cdots 75}a^{2}-\frac{87\cdots 49}{25\cdots 95}a-\frac{67\cdots 97}{50\cdots 19}$, $\frac{19\cdots 57}{14\cdots 75}a^{17}-\frac{72\cdots 03}{14\cdots 75}a^{16}-\frac{78\cdots 01}{14\cdots 75}a^{15}+\frac{29\cdots 44}{28\cdots 55}a^{14}+\frac{33\cdots 41}{28\cdots 55}a^{13}+\frac{30\cdots 73}{14\cdots 75}a^{12}-\frac{24\cdots 01}{20\cdots 25}a^{11}-\frac{31\cdots 93}{14\cdots 75}a^{10}+\frac{56\cdots 93}{14\cdots 75}a^{9}+\frac{26\cdots 47}{14\cdots 75}a^{8}+\frac{53\cdots 89}{28\cdots 55}a^{7}-\frac{37\cdots 99}{14\cdots 75}a^{6}-\frac{14\cdots 73}{14\cdots 75}a^{5}-\frac{20\cdots 01}{14\cdots 75}a^{4}-\frac{67\cdots 00}{56\cdots 91}a^{3}-\frac{91\cdots 51}{14\cdots 75}a^{2}-\frac{62\cdots 01}{28\cdots 55}a-\frac{25\cdots 81}{56\cdots 91}$, $\frac{27\cdots 08}{20\cdots 25}a^{17}-\frac{50\cdots 87}{42\cdots 25}a^{16}-\frac{14\cdots 39}{42\cdots 25}a^{15}+\frac{71\cdots 68}{16\cdots 73}a^{14}+\frac{22\cdots 73}{28\cdots 55}a^{13}-\frac{39\cdots 39}{60\cdots 75}a^{12}-\frac{68\cdots 03}{42\cdots 25}a^{11}+\frac{20\cdots 83}{42\cdots 25}a^{10}+\frac{38\cdots 27}{20\cdots 25}a^{9}-\frac{19\cdots 82}{42\cdots 25}a^{8}-\frac{87\cdots 22}{84\cdots 65}a^{7}-\frac{58\cdots 01}{42\cdots 25}a^{6}+\frac{63\cdots 79}{60\cdots 75}a^{5}+\frac{21\cdots 86}{42\cdots 25}a^{4}+\frac{53\cdots 34}{84\cdots 65}a^{3}+\frac{17\cdots 26}{42\cdots 25}a^{2}+\frac{96\cdots 74}{56\cdots 91}a+\frac{67\cdots 83}{16\cdots 73}$ 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:  \( 625949725971 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{7}\cdot 625949725971 \cdot 1}{2\cdot\sqrt{14616700658977641856215848806693687375}}\cr\approx \mathstrut & 0.506365190150983 \end{aligned}\] (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 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125) 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 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125, 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 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125); 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 = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 - 23*x^16 + 300*x^15 + 515*x^14 - 4561*x^13 - 10821*x^12 + 31931*x^11 + 128094*x^10 - 8849*x^9 - 656070*x^8 - 1052332*x^7 + 297296*x^6 + 3208702*x^5 + 5247365*x^4 + 4745832*x^3 + 2696725*x^2 + 963650*x + 201125); 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

$C_3^6.C_2\wr C_6$ (as 18T858):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 279936
The 159 conjugacy class representatives for $C_3^6.C_2\wr C_6$
Character table for $C_3^6.C_2\wr C_6$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.1006019.1

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(L)]
 
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 24 sibling: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.8.3838671405368745874099609375.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{4}$ R R $18$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ $18$ ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ $18$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ R ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$

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
\(5\) Copy content Toggle raw display 5.3.1.0a1.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.9.1.0a1.1$x^{9} + 2 x^{3} + x + 3$$1$$9$$0$$C_9$$$[\ ]^{9}$$
\(7\) Copy content Toggle raw display 7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.2.3.4a1.2$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 162 x + 34$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$
\(41\) Copy content Toggle raw display 41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.3.1.0a1.1$x^{3} + x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
41.3.1.0a1.1$x^{3} + x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
41.2.3.4a1.1$x^{6} + 114 x^{5} + 4350 x^{4} + 56240 x^{3} + 26100 x^{2} + 4145 x + 216$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(419\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$2$$3$$3$
\(449\) Copy content Toggle raw display Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$3$$2$$4$

Spectrum of ring of integers

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