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Properties

Label 18.10.125...184.3
Degree $18$
Signature $[10, 4]$
Discriminant $1.257\times 10^{24}$
Root discriminant \(21.82\)
Ramified primes $2,7,53$
Class number $1$
Class group trivial
Galois group $C_2^4:(S_3\times A_4)$ (as 18T268)

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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 + 2*x^16 - 32*x^14 + 27*x^12 + 147*x^10 - 229*x^8 + 27*x^6 + 34*x^4 - 3*x^2 - 1)
 
Copy content gp:K = bnfinit(y^18 + 2*y^16 - 32*y^14 + 27*y^12 + 147*y^10 - 229*y^8 + 27*y^6 + 34*y^4 - 3*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 2*x^16 - 32*x^14 + 27*x^12 + 147*x^10 - 229*x^8 + 27*x^6 + 34*x^4 - 3*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 2*x^16 - 32*x^14 + 27*x^12 + 147*x^10 - 229*x^8 + 27*x^6 + 34*x^4 - 3*x^2 - 1)
 

\( x^{18} + 2x^{16} - 32x^{14} + 27x^{12} + 147x^{10} - 229x^{8} + 27x^{6} + 34x^{4} - 3x^{2} - 1 \) 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:  $[10, 4]$
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:   \(1256584347701942722269184\) \(\medspace = 2^{12}\cdot 7^{12}\cdot 53^{6}\) 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:  \(21.82\)
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:  $2^{15/8}7^{2/3}53^{1/2}\approx 97.71649256380418$
Ramified primes:   \(2\), \(7\), \(53\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{582}a^{14}+\frac{41}{291}a^{12}-\frac{115}{582}a^{10}-\frac{21}{97}a^{8}-\frac{1}{2}a^{7}+\frac{31}{582}a^{6}-\frac{1}{2}a^{5}+\frac{25}{582}a^{4}+\frac{85}{582}a^{2}-\frac{1}{2}a+\frac{227}{582}$, $\frac{1}{582}a^{15}+\frac{41}{291}a^{13}-\frac{115}{582}a^{11}-\frac{21}{97}a^{9}-\frac{1}{2}a^{8}+\frac{31}{582}a^{7}-\frac{1}{2}a^{6}+\frac{25}{582}a^{5}+\frac{85}{582}a^{3}-\frac{1}{2}a^{2}+\frac{227}{582}a$, $\frac{1}{1746}a^{16}+\frac{218}{873}a^{12}-\frac{4}{873}a^{10}-\frac{202}{873}a^{8}+\frac{17}{291}a^{6}-\frac{85}{291}a^{4}-\frac{341}{1746}a^{2}-\frac{286}{873}$, $\frac{1}{1746}a^{17}+\frac{218}{873}a^{13}-\frac{4}{873}a^{11}-\frac{202}{873}a^{9}+\frac{17}{291}a^{7}-\frac{85}{291}a^{5}-\frac{341}{1746}a^{3}-\frac{286}{873}a$ 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:  Trivial group, which has order $1$
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:  $C_{2}$, which has order $2$
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:  $13$
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:   $\frac{92}{873}a^{16}+\frac{103}{291}a^{14}-\frac{2644}{873}a^{12}-\frac{1351}{873}a^{10}+\frac{15563}{873}a^{8}-\frac{318}{97}a^{6}-\frac{2221}{97}a^{4}+\frac{1877}{873}a^{2}+\frac{2678}{873}$, $\frac{107}{873}a^{16}+\frac{16}{97}a^{14}-\frac{3523}{873}a^{12}+\frac{5282}{873}a^{10}+\frac{12833}{873}a^{8}-\frac{11752}{291}a^{6}+\frac{7745}{291}a^{4}-\frac{676}{873}a^{2}-\frac{2326}{873}$, $a$, $\frac{14}{97}a^{17}+\frac{1}{6}a^{15}-\frac{2855}{582}a^{13}+\frac{4469}{582}a^{11}+\frac{3723}{194}a^{9}-\frac{29747}{582}a^{7}+\frac{14875}{582}a^{5}+\frac{4045}{582}a^{3}-\frac{938}{291}a$, $\frac{1003}{1746}a^{17}+\frac{325}{291}a^{15}-\frac{32225}{1746}a^{13}+\frac{28753}{1746}a^{11}+\frac{146137}{1746}a^{9}-\frac{26245}{194}a^{7}+\frac{4355}{194}a^{5}+\frac{12695}{873}a^{3}-\frac{989}{1746}a$, $\frac{71}{582}a^{16}+\frac{185}{582}a^{14}-\frac{1090}{291}a^{12}+\frac{94}{97}a^{10}+\frac{5722}{291}a^{8}-\frac{5006}{291}a^{6}-\frac{3425}{291}a^{4}+\frac{2863}{582}a^{2}+\frac{73}{194}$, $\frac{5}{873}a^{16}-\frac{49}{582}a^{14}-\frac{355}{873}a^{12}+\frac{2738}{873}a^{10}-\frac{616}{873}a^{8}-\frac{1506}{97}a^{6}+\frac{1598}{97}a^{4}+\frac{1214}{873}a^{2}-\frac{677}{1746}$, $\frac{161}{873}a^{17}+\frac{76}{873}a^{16}+\frac{65}{291}a^{15}+\frac{185}{582}a^{14}-\frac{5479}{873}a^{13}-\frac{4327}{1746}a^{12}+\frac{16303}{1746}a^{11}-\frac{3931}{1746}a^{10}+\frac{22130}{873}a^{9}+\frac{142}{9}a^{8}-\frac{36529}{582}a^{7}+\frac{650}{291}a^{6}+\frac{16313}{582}a^{5}-\frac{14813}{582}a^{4}+\frac{13267}{1746}a^{3}+\frac{727}{873}a^{2}-\frac{1558}{873}a+\frac{1624}{873}$, $\frac{19}{97}a^{16}+\frac{277}{582}a^{14}-\frac{3533}{582}a^{12}+\frac{1571}{582}a^{10}+\frac{5897}{194}a^{8}-\frac{18779}{582}a^{6}-\frac{6401}{582}a^{4}+\frac{2713}{582}a^{2}+\frac{436}{291}$, $\frac{641}{1746}a^{17}-\frac{5}{1746}a^{16}+\frac{199}{291}a^{15}+\frac{16}{291}a^{14}-\frac{10352}{873}a^{13}+\frac{227}{873}a^{12}+\frac{9970}{873}a^{11}-\frac{3143}{1746}a^{10}+\frac{45847}{873}a^{9}+\frac{200}{873}a^{8}-\frac{52295}{582}a^{7}+\frac{1535}{194}a^{6}+\frac{13015}{582}a^{5}-\frac{1875}{194}a^{4}+\frac{8621}{1746}a^{3}+\frac{2750}{873}a^{2}-\frac{2203}{1746}a+\frac{104}{873}$, $\frac{161}{873}a^{17}+\frac{103}{291}a^{16}+\frac{65}{291}a^{15}+\frac{63}{97}a^{14}-\frac{5479}{873}a^{13}-\frac{3323}{291}a^{12}+\frac{16303}{1746}a^{11}+\frac{3340}{291}a^{10}+\frac{22130}{873}a^{9}+\frac{28907}{582}a^{8}-\frac{36529}{582}a^{7}-\frac{17317}{194}a^{6}+\frac{16313}{582}a^{5}+\frac{2592}{97}a^{4}+\frac{13267}{1746}a^{3}+\frac{4079}{582}a^{2}-\frac{1558}{873}a-\frac{881}{291}$, $\frac{20}{873}a^{17}-\frac{79}{291}a^{16}+\frac{61}{291}a^{15}-\frac{105}{194}a^{14}-\frac{563}{1746}a^{13}+\frac{5095}{582}a^{12}-\frac{3745}{873}a^{11}-\frac{2062}{291}a^{10}+\frac{10183}{1746}a^{9}-\frac{11968}{291}a^{8}+\frac{3363}{194}a^{7}+\frac{11743}{194}a^{6}-\frac{5421}{194}a^{5}-\frac{250}{97}a^{4}+\frac{1751}{873}a^{3}-\frac{4907}{582}a^{2}+\frac{2165}{873}a-\frac{22}{291}$, $\frac{17}{291}a^{17}-\frac{47}{1746}a^{16}+\frac{9}{97}a^{15}-\frac{13}{582}a^{14}-\frac{559}{291}a^{13}+\frac{1627}{1746}a^{12}+\frac{1375}{582}a^{11}-\frac{1498}{873}a^{10}+\frac{4777}{582}a^{9}-\frac{2890}{873}a^{8}-\frac{1665}{97}a^{7}+\frac{1073}{97}a^{6}+\frac{975}{194}a^{5}-\frac{1325}{194}a^{4}+\frac{572}{291}a^{3}-\frac{2374}{873}a^{2}+\frac{479}{291}a+\frac{571}{1746}$ Copy content Toggle raw display
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:  \( 289447.222908 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 289447.222908 \cdot 1}{2\cdot\sqrt{1256584347701942722269184}}\cr\approx \mathstrut & 0.206045435757 \end{aligned}\]

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 + 2*x^16 - 32*x^14 + 27*x^12 + 147*x^10 - 229*x^8 + 27*x^6 + 34*x^4 - 3*x^2 - 1) 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 + 2*x^16 - 32*x^14 + 27*x^12 + 147*x^10 - 229*x^8 + 27*x^6 + 34*x^4 - 3*x^2 - 1, 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 + 2*x^16 - 32*x^14 + 27*x^12 + 147*x^10 - 229*x^8 + 27*x^6 + 34*x^4 - 3*x^2 - 1); 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 + 2*x^16 - 32*x^14 + 27*x^12 + 147*x^10 - 229*x^8 + 27*x^6 + 34*x^4 - 3*x^2 - 1); 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

$C_2^4:(S_3\times A_4)$ (as 18T268):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 1152
The 24 conjugacy class representatives for $C_2^4:(S_3\times A_4)$
Character table for $C_2^4:(S_3\times A_4)$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.4.186314968102014976.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ R ${\href{/padicField/59.3.0.1}{3} }^{6}$

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 2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.6.2.12a11.1$x^{12} + 2 x^{11} + 4 x^{10} + 4 x^{9} + 7 x^{8} + 8 x^{7} + 7 x^{6} + 10 x^{5} + 8 x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 3$$2$$6$$12$12T58$$[2, 2, 2, 2]^{6}$$
\(7\) Copy content Toggle raw display 7.3.3.6a1.3$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$
7.3.3.6a1.3$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$
\(53\) Copy content Toggle raw display 53.3.1.0a1.1$x^{3} + 3 x + 51$$1$$3$$0$$C_3$$$[\ ]^{3}$$
53.3.1.0a1.1$x^{3} + 3 x + 51$$1$$3$$0$$C_3$$$[\ ]^{3}$$
53.3.2.3a1.2$x^{6} + 6 x^{4} + 102 x^{3} + 9 x^{2} + 306 x + 2654$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
53.3.2.3a1.2$x^{6} + 6 x^{4} + 102 x^{3} + 9 x^{2} + 306 x + 2654$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Spectrum of ring of integers

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