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Properties

Label 18.18.239...952.2
Degree $18$
Signature $[18, 0]$
Discriminant $2.397\times 10^{36}$
Root discriminant \(104.98\)
Ramified primes $2,257,43237$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $S_4^3.S_4$ (as 18T884)

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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 - 103*x^16 + 4170*x^14 - 88068*x^12 + 1065425*x^10 - 7547378*x^8 + 30446048*x^6 - 63809759*x^4 + 56537944*x^2 - 16974593)
 
Copy content gp:K = bnfinit(y^18 - 103*y^16 + 4170*y^14 - 88068*y^12 + 1065425*y^10 - 7547378*y^8 + 30446048*y^6 - 63809759*y^4 + 56537944*y^2 - 16974593, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 103*x^16 + 4170*x^14 - 88068*x^12 + 1065425*x^10 - 7547378*x^8 + 30446048*x^6 - 63809759*x^4 + 56537944*x^2 - 16974593);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 103*x^16 + 4170*x^14 - 88068*x^12 + 1065425*x^10 - 7547378*x^8 + 30446048*x^6 - 63809759*x^4 + 56537944*x^2 - 16974593)
 

\( x^{18} - 103 x^{16} + 4170 x^{14} - 88068 x^{12} + 1065425 x^{10} - 7547378 x^{8} + 30446048 x^{6} + \cdots - 16974593 \) 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:  $[18, 0]$
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:   \(2396895706841866493030330389189885952\) \(\medspace = 2^{18}\cdot 257^{9}\cdot 43237^{2}\) 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:  \(104.98\)
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:  $2^{63/32}257^{1/2}43237^{1/2}\approx 13048.095961157862$
Ramified primes:   \(2\), \(257\), \(43237\) 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:  \(\Q(\sqrt{257}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{257}a^{14}-\frac{103}{257}a^{12}+\frac{58}{257}a^{10}+\frac{83}{257}a^{8}-\frac{97}{257}a^{6}-\frac{59}{257}a^{4}+\frac{29}{257}a^{2}$, $\frac{1}{257}a^{15}-\frac{103}{257}a^{13}+\frac{58}{257}a^{11}+\frac{83}{257}a^{9}-\frac{97}{257}a^{7}-\frac{59}{257}a^{5}+\frac{29}{257}a^{3}$, $\frac{1}{63\cdots 89}a^{16}-\frac{95\cdots 31}{63\cdots 89}a^{14}-\frac{14\cdots 46}{63\cdots 89}a^{12}-\frac{23\cdots 34}{63\cdots 89}a^{10}+\frac{14\cdots 89}{63\cdots 89}a^{8}+\frac{26\cdots 80}{63\cdots 89}a^{6}-\frac{60\cdots 32}{63\cdots 89}a^{4}+\frac{42\cdots 52}{24\cdots 77}a^{2}+\frac{31\cdots 89}{96\cdots 61}$, $\frac{1}{63\cdots 89}a^{17}-\frac{95\cdots 31}{63\cdots 89}a^{15}-\frac{14\cdots 46}{63\cdots 89}a^{13}-\frac{23\cdots 34}{63\cdots 89}a^{11}+\frac{14\cdots 89}{63\cdots 89}a^{9}+\frac{26\cdots 80}{63\cdots 89}a^{7}-\frac{60\cdots 32}{63\cdots 89}a^{5}+\frac{42\cdots 52}{24\cdots 77}a^{3}+\frac{31\cdots 89}{96\cdots 61}a$ 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:  $C_{2}$, which has order $2$ (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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $32$ (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:  $17$
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:   $\frac{20508331589240}{24\cdots 77}a^{16}-\frac{19\cdots 50}{24\cdots 77}a^{14}+\frac{66\cdots 91}{24\cdots 77}a^{12}-\frac{11\cdots 51}{24\cdots 77}a^{10}+\frac{11\cdots 90}{24\cdots 77}a^{8}-\frac{57\cdots 47}{24\cdots 77}a^{6}+\frac{57\cdots 10}{96\cdots 61}a^{4}-\frac{14\cdots 81}{24\cdots 77}a^{2}+\frac{17\cdots 51}{96\cdots 61}$, $\frac{37\cdots 56}{63\cdots 89}a^{16}-\frac{36\cdots 29}{63\cdots 89}a^{14}+\frac{13\cdots 11}{63\cdots 89}a^{12}-\frac{25\cdots 08}{63\cdots 89}a^{10}+\frac{25\cdots 81}{63\cdots 89}a^{8}-\frac{14\cdots 42}{63\cdots 89}a^{6}+\frac{42\cdots 57}{63\cdots 89}a^{4}-\frac{22\cdots 05}{24\cdots 77}a^{2}+\frac{41\cdots 59}{96\cdots 61}$, $\frac{31\cdots 93}{63\cdots 89}a^{16}-\frac{25\cdots 12}{63\cdots 89}a^{14}+\frac{67\cdots 55}{63\cdots 89}a^{12}-\frac{78\cdots 02}{63\cdots 89}a^{10}+\frac{35\cdots 53}{63\cdots 89}a^{8}-\frac{13\cdots 98}{63\cdots 89}a^{6}-\frac{28\cdots 62}{63\cdots 89}a^{4}+\frac{18\cdots 60}{24\cdots 77}a^{2}-\frac{37\cdots 21}{96\cdots 61}$, $\frac{12\cdots 84}{63\cdots 89}a^{16}-\frac{11\cdots 70}{63\cdots 89}a^{14}+\frac{41\cdots 42}{63\cdots 89}a^{12}-\frac{75\cdots 53}{63\cdots 89}a^{10}+\frac{73\cdots 87}{63\cdots 89}a^{8}-\frac{37\cdots 11}{63\cdots 89}a^{6}+\frac{90\cdots 79}{63\cdots 89}a^{4}-\frac{29\cdots 75}{24\cdots 77}a^{2}+\frac{26\cdots 04}{96\cdots 61}$, $\frac{33605876764195}{24\cdots 77}a^{16}-\frac{30\cdots 38}{24\cdots 77}a^{14}+\frac{10\cdots 83}{24\cdots 77}a^{12}-\frac{16\cdots 87}{24\cdots 77}a^{10}+\frac{15\cdots 67}{24\cdots 77}a^{8}-\frac{75\cdots 87}{24\cdots 77}a^{6}+\frac{73\cdots 36}{96\cdots 61}a^{4}-\frac{20\cdots 64}{24\cdots 77}a^{2}+\frac{32\cdots 40}{96\cdots 61}$, $\frac{126636787985}{96\cdots 61}a^{16}-\frac{12865335851708}{96\cdots 61}a^{14}+\frac{501208244441343}{96\cdots 61}a^{12}-\frac{97\cdots 83}{96\cdots 61}a^{10}+\frac{10\cdots 37}{96\cdots 61}a^{8}-\frac{55\cdots 29}{96\cdots 61}a^{6}+\frac{14\cdots 86}{96\cdots 61}a^{4}-\frac{15\cdots 49}{96\cdots 61}a^{2}+\frac{56\cdots 73}{96\cdots 61}$, $\frac{11\cdots 00}{63\cdots 89}a^{16}-\frac{10\cdots 59}{63\cdots 89}a^{14}+\frac{38\cdots 59}{63\cdots 89}a^{12}-\frac{67\cdots 79}{63\cdots 89}a^{10}+\frac{64\cdots 24}{63\cdots 89}a^{8}-\frac{33\cdots 85}{63\cdots 89}a^{6}+\frac{88\cdots 22}{63\cdots 89}a^{4}-\frac{39\cdots 39}{24\cdots 77}a^{2}+\frac{55\cdots 52}{96\cdots 61}$, $\frac{65\cdots 20}{63\cdots 89}a^{16}-\frac{60\cdots 09}{63\cdots 89}a^{14}+\frac{21\cdots 72}{63\cdots 89}a^{12}-\frac{37\cdots 72}{63\cdots 89}a^{10}+\frac{35\cdots 94}{63\cdots 89}a^{8}-\frac{18\cdots 06}{63\cdots 89}a^{6}+\frac{50\cdots 32}{63\cdots 89}a^{4}-\frac{24\cdots 58}{24\cdots 77}a^{2}+\frac{37\cdots 40}{96\cdots 61}$, $\frac{17\cdots 06}{63\cdots 89}a^{16}-\frac{16\cdots 24}{63\cdots 89}a^{14}+\frac{61\cdots 18}{63\cdots 89}a^{12}-\frac{11\cdots 10}{63\cdots 89}a^{10}+\frac{10\cdots 53}{63\cdots 89}a^{8}-\frac{56\cdots 72}{63\cdots 89}a^{6}+\frac{14\cdots 05}{63\cdots 89}a^{4}-\frac{52\cdots 99}{24\cdots 77}a^{2}+a+\frac{61\cdots 04}{96\cdots 61}$, $\frac{68\cdots 29}{63\cdots 89}a^{17}-\frac{28\cdots 96}{63\cdots 89}a^{16}-\frac{62\cdots 07}{63\cdots 89}a^{15}+\frac{24\cdots 67}{63\cdots 89}a^{14}+\frac{21\cdots 17}{63\cdots 89}a^{13}-\frac{72\cdots 13}{63\cdots 89}a^{12}-\frac{35\cdots 92}{63\cdots 89}a^{11}+\frac{10\cdots 01}{63\cdots 89}a^{10}+\frac{31\cdots 60}{63\cdots 89}a^{9}-\frac{84\cdots 34}{63\cdots 89}a^{8}-\frac{14\cdots 01}{63\cdots 89}a^{7}+\frac{40\cdots 57}{63\cdots 89}a^{6}+\frac{28\cdots 64}{63\cdots 89}a^{5}-\frac{12\cdots 29}{63\cdots 89}a^{4}-\frac{49\cdots 91}{24\cdots 77}a^{3}+\frac{93\cdots 20}{24\cdots 77}a^{2}-\frac{65\cdots 77}{96\cdots 61}a-\frac{21\cdots 78}{96\cdots 61}$, $\frac{30\cdots 08}{63\cdots 89}a^{17}-\frac{41\cdots 11}{63\cdots 89}a^{16}-\frac{30\cdots 53}{63\cdots 89}a^{15}+\frac{41\cdots 58}{63\cdots 89}a^{14}+\frac{11\cdots 23}{63\cdots 89}a^{13}-\frac{15\cdots 53}{63\cdots 89}a^{12}-\frac{23\cdots 68}{63\cdots 89}a^{11}+\frac{29\cdots 02}{63\cdots 89}a^{10}+\frac{25\cdots 97}{63\cdots 89}a^{9}-\frac{31\cdots 40}{63\cdots 89}a^{8}-\frac{15\cdots 92}{63\cdots 89}a^{7}+\frac{18\cdots 83}{63\cdots 89}a^{6}+\frac{52\cdots 39}{63\cdots 89}a^{5}-\frac{56\cdots 32}{63\cdots 89}a^{4}-\frac{32\cdots 00}{24\cdots 77}a^{3}+\frac{31\cdots 54}{24\cdots 77}a^{2}+\frac{65\cdots 78}{96\cdots 61}a-\frac{58\cdots 59}{96\cdots 61}$, $\frac{47\cdots 44}{63\cdots 89}a^{17}-\frac{15\cdots 71}{63\cdots 89}a^{16}-\frac{46\cdots 34}{63\cdots 89}a^{15}+\frac{14\cdots 85}{63\cdots 89}a^{14}+\frac{16\cdots 26}{63\cdots 89}a^{13}-\frac{52\cdots 66}{63\cdots 89}a^{12}-\frac{31\cdots 05}{63\cdots 89}a^{11}+\frac{94\cdots 53}{63\cdots 89}a^{10}+\frac{30\cdots 45}{63\cdots 89}a^{9}-\frac{91\cdots 23}{63\cdots 89}a^{8}-\frac{15\cdots 87}{63\cdots 89}a^{7}+\frac{47\cdots 17}{63\cdots 89}a^{6}+\frac{38\cdots 32}{63\cdots 89}a^{5}-\frac{11\cdots 03}{63\cdots 89}a^{4}-\frac{13\cdots 33}{24\cdots 77}a^{3}+\frac{44\cdots 69}{24\cdots 77}a^{2}+\frac{14\cdots 65}{96\cdots 61}a-\frac{53\cdots 76}{96\cdots 61}$, $\frac{106643983777602}{24\cdots 77}a^{17}-\frac{32\cdots 39}{63\cdots 89}a^{16}-\frac{96\cdots 13}{24\cdots 77}a^{15}+\frac{26\cdots 00}{63\cdots 89}a^{14}+\frac{32\cdots 17}{24\cdots 77}a^{13}-\frac{78\cdots 64}{63\cdots 89}a^{12}-\frac{55\cdots 42}{24\cdots 77}a^{11}+\frac{10\cdots 42}{63\cdots 89}a^{10}+\frac{49\cdots 90}{24\cdots 77}a^{9}-\frac{67\cdots 06}{63\cdots 89}a^{8}-\frac{23\cdots 88}{24\cdots 77}a^{7}+\frac{19\cdots 98}{63\cdots 89}a^{6}+\frac{21\cdots 52}{96\cdots 61}a^{5}-\frac{18\cdots 22}{63\cdots 89}a^{4}-\frac{49\cdots 66}{24\cdots 77}a^{3}-\frac{15\cdots 41}{24\cdots 77}a^{2}+\frac{56\cdots 39}{96\cdots 61}a+\frac{10\cdots 18}{96\cdots 61}$, $\frac{69\cdots 30}{63\cdots 89}a^{17}-\frac{16\cdots 17}{63\cdots 89}a^{16}-\frac{66\cdots 01}{63\cdots 89}a^{15}+\frac{15\cdots 69}{63\cdots 89}a^{14}+\frac{24\cdots 12}{63\cdots 89}a^{13}-\frac{58\cdots 32}{63\cdots 89}a^{12}-\frac{44\cdots 42}{63\cdots 89}a^{11}+\frac{10\cdots 48}{63\cdots 89}a^{10}+\frac{43\cdots 20}{63\cdots 89}a^{9}-\frac{10\cdots 05}{63\cdots 89}a^{8}-\frac{22\cdots 30}{63\cdots 89}a^{7}+\frac{56\cdots 77}{63\cdots 89}a^{6}+\frac{56\cdots 00}{63\cdots 89}a^{5}-\frac{14\cdots 40}{63\cdots 89}a^{4}-\frac{21\cdots 24}{24\cdots 77}a^{3}+\frac{55\cdots 05}{24\cdots 77}a^{2}+\frac{26\cdots 26}{96\cdots 61}a-\frac{67\cdots 06}{96\cdots 61}$, $\frac{18\cdots 20}{63\cdots 89}a^{17}-\frac{13\cdots 37}{63\cdots 89}a^{16}-\frac{17\cdots 33}{63\cdots 89}a^{15}+\frac{13\cdots 56}{63\cdots 89}a^{14}+\frac{62\cdots 55}{63\cdots 89}a^{13}-\frac{49\cdots 72}{63\cdots 89}a^{12}-\frac{11\cdots 42}{63\cdots 89}a^{11}+\frac{88\cdots 79}{63\cdots 89}a^{10}+\frac{11\cdots 88}{63\cdots 89}a^{9}-\frac{83\cdots 62}{63\cdots 89}a^{8}-\frac{61\cdots 66}{63\cdots 89}a^{7}+\frac{39\cdots 39}{63\cdots 89}a^{6}+\frac{17\cdots 67}{63\cdots 89}a^{5}-\frac{81\cdots 94}{63\cdots 89}a^{4}-\frac{81\cdots 00}{24\cdots 77}a^{3}+\frac{14\cdots 32}{24\cdots 77}a^{2}+\frac{12\cdots 34}{96\cdots 61}a+\frac{33\cdots 77}{96\cdots 61}$, $\frac{13\cdots 83}{63\cdots 89}a^{17}+\frac{19\cdots 03}{63\cdots 89}a^{16}-\frac{12\cdots 52}{63\cdots 89}a^{15}-\frac{18\cdots 53}{63\cdots 89}a^{14}+\frac{46\cdots 59}{63\cdots 89}a^{13}+\frac{64\cdots 26}{63\cdots 89}a^{12}-\frac{83\cdots 92}{63\cdots 89}a^{11}-\frac{11\cdots 88}{63\cdots 89}a^{10}+\frac{81\cdots 64}{63\cdots 89}a^{9}+\frac{10\cdots 12}{63\cdots 89}a^{8}-\frac{43\cdots 01}{63\cdots 89}a^{7}-\frac{55\cdots 74}{63\cdots 89}a^{6}+\frac{11\cdots 86}{63\cdots 89}a^{5}+\frac{14\cdots 98}{63\cdots 89}a^{4}-\frac{50\cdots 67}{24\cdots 77}a^{3}-\frac{63\cdots 41}{24\cdots 77}a^{2}+\frac{69\cdots 84}{96\cdots 61}a+\frac{88\cdots 58}{96\cdots 61}$, $\frac{23\cdots 34}{63\cdots 89}a^{17}+\frac{85\cdots 79}{63\cdots 89}a^{16}-\frac{20\cdots 86}{63\cdots 89}a^{15}-\frac{75\cdots 04}{63\cdots 89}a^{14}+\frac{68\cdots 28}{63\cdots 89}a^{13}+\frac{24\cdots 34}{63\cdots 89}a^{12}-\frac{11\cdots 01}{63\cdots 89}a^{11}-\frac{39\cdots 22}{63\cdots 89}a^{10}+\frac{96\cdots 67}{63\cdots 89}a^{9}+\frac{33\cdots 94}{63\cdots 89}a^{8}-\frac{45\cdots 59}{63\cdots 89}a^{7}-\frac{15\cdots 42}{63\cdots 89}a^{6}+\frac{10\cdots 92}{63\cdots 89}a^{5}+\frac{35\cdots 70}{63\cdots 89}a^{4}-\frac{39\cdots 72}{24\cdots 77}a^{3}-\frac{12\cdots 94}{24\cdots 77}a^{2}+\frac{53\cdots 32}{96\cdots 61}a+\frac{14\cdots 87}{96\cdots 61}$ 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:  \( 1174071788510 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 1174071788510 \cdot 2}{2\cdot\sqrt{2396895706841866493030330389189885952}}\cr\approx \mathstrut & 0.198797082181842 \end{aligned}\] (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 - 103*x^16 + 4170*x^14 - 88068*x^12 + 1065425*x^10 - 7547378*x^8 + 30446048*x^6 - 63809759*x^4 + 56537944*x^2 - 16974593) 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 - 103*x^16 + 4170*x^14 - 88068*x^12 + 1065425*x^10 - 7547378*x^8 + 30446048*x^6 - 63809759*x^4 + 56537944*x^2 - 16974593, 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 - 103*x^16 + 4170*x^14 - 88068*x^12 + 1065425*x^10 - 7547378*x^8 + 30446048*x^6 - 63809759*x^4 + 56537944*x^2 - 16974593); 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 - 103*x^16 + 4170*x^14 - 88068*x^12 + 1065425*x^10 - 7547378*x^8 + 30446048*x^6 - 63809759*x^4 + 56537944*x^2 - 16974593); 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

$S_4^3.S_4$ (as 18T884):

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 331776
The 165 conjugacy class representatives for $S_4^3.S_4$
Character table for $S_4^3.S_4$

Intermediate fields

3.3.257.1, 9.9.733930477541.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.18.103634579676721781559152395037403098906624.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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

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 2.3.2.6a1.1$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$$2$$3$$6$$C_6$$$[2]^{3}$$
2.6.2.12a8.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(257\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
\(43237\) Copy content Toggle raw display $\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $6$$1$$6$$0$$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)