Properties

Label 18.6.458...249.1
Degree $18$
Signature $[6, 6]$
Discriminant $4.581\times 10^{24}$
Root discriminant \(23.45\)
Ramified primes $7,13$
Class number $1$
Class group trivial
Galois group $C_3^2.A_4$ (as 18T47)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 4*x^16 + 12*x^15 - 20*x^14 + 28*x^13 + 84*x^12 - 232*x^11 - 180*x^10 + 900*x^9 - 334*x^8 - 1660*x^7 + 2996*x^6 - 2576*x^5 + 1380*x^4 - 492*x^3 + 115*x^2 - 16*x + 1)
 
gp: K = bnfinit(y^18 - 2*y^17 - 4*y^16 + 12*y^15 - 20*y^14 + 28*y^13 + 84*y^12 - 232*y^11 - 180*y^10 + 900*y^9 - 334*y^8 - 1660*y^7 + 2996*y^6 - 2576*y^5 + 1380*y^4 - 492*y^3 + 115*y^2 - 16*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 - 4*x^16 + 12*x^15 - 20*x^14 + 28*x^13 + 84*x^12 - 232*x^11 - 180*x^10 + 900*x^9 - 334*x^8 - 1660*x^7 + 2996*x^6 - 2576*x^5 + 1380*x^4 - 492*x^3 + 115*x^2 - 16*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 - 4*x^16 + 12*x^15 - 20*x^14 + 28*x^13 + 84*x^12 - 232*x^11 - 180*x^10 + 900*x^9 - 334*x^8 - 1660*x^7 + 2996*x^6 - 2576*x^5 + 1380*x^4 - 492*x^3 + 115*x^2 - 16*x + 1)
 

\( x^{18} - 2 x^{17} - 4 x^{16} + 12 x^{15} - 20 x^{14} + 28 x^{13} + 84 x^{12} - 232 x^{11} - 180 x^{10} + 900 x^{9} - 334 x^{8} - 1660 x^{7} + 2996 x^{6} - 2576 x^{5} + 1380 x^{4} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[6, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4581441688047722385682249\) \(\medspace = 7^{16}\cdot 13^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(23.45\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $7^{8/9}13^{5/6}\approx 47.806217686235044$
Ramified primes:   \(7\), \(13\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{6}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{6}a^{11}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}$, $\frac{1}{6}a^{13}-\frac{1}{3}a^{7}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}$, $\frac{1}{42}a^{14}-\frac{1}{14}a^{13}+\frac{1}{14}a^{12}+\frac{1}{21}a^{10}-\frac{1}{42}a^{9}+\frac{5}{21}a^{8}+\frac{2}{7}a^{7}-\frac{19}{42}a^{6}-\frac{1}{21}a^{5}+\frac{5}{14}a^{3}-\frac{2}{21}a^{2}+\frac{5}{42}a-\frac{17}{42}$, $\frac{1}{42}a^{15}+\frac{1}{42}a^{13}+\frac{1}{21}a^{12}+\frac{1}{21}a^{11}-\frac{1}{21}a^{10}+\frac{1}{3}a^{8}-\frac{11}{42}a^{7}-\frac{17}{42}a^{6}-\frac{13}{42}a^{5}+\frac{1}{42}a^{4}+\frac{1}{7}a^{3}-\frac{1}{3}a^{2}-\frac{1}{21}a-\frac{8}{21}$, $\frac{1}{42}a^{16}-\frac{1}{21}a^{13}-\frac{1}{42}a^{12}-\frac{1}{21}a^{11}-\frac{1}{21}a^{10}+\frac{1}{42}a^{9}-\frac{1}{6}a^{8}-\frac{1}{42}a^{7}+\frac{10}{21}a^{6}-\frac{3}{7}a^{5}+\frac{13}{42}a^{4}+\frac{13}{42}a^{3}+\frac{1}{21}a^{2}+\frac{1}{6}a-\frac{11}{42}$, $\frac{1}{42}a^{17}-\frac{1}{14}a^{12}-\frac{1}{21}a^{11}-\frac{1}{21}a^{10}-\frac{1}{21}a^{9}+\frac{19}{42}a^{8}+\frac{1}{21}a^{7}+\frac{1}{3}a^{6}-\frac{2}{7}a^{5}-\frac{1}{42}a^{4}-\frac{1}{14}a^{3}-\frac{4}{21}a^{2}+\frac{13}{42}a+\frac{5}{14}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{188}{21}a^{17}-\frac{274}{21}a^{16}-\frac{1825}{42}a^{15}+\frac{1775}{21}a^{14}-\frac{913}{7}a^{13}+\frac{2455}{14}a^{12}+\frac{17954}{21}a^{11}-\frac{34073}{21}a^{10}-\frac{53476}{21}a^{9}+6762a^{8}+\frac{35527}{42}a^{7}-\frac{620531}{42}a^{6}+\frac{392870}{21}a^{5}-\frac{506395}{42}a^{4}+\frac{28703}{6}a^{3}-\frac{24398}{21}a^{2}+\frac{3134}{21}a-\frac{130}{21}$, $\frac{188}{21}a^{17}-\frac{274}{21}a^{16}-\frac{1825}{42}a^{15}+\frac{1775}{21}a^{14}-\frac{913}{7}a^{13}+\frac{2455}{14}a^{12}+\frac{17954}{21}a^{11}-\frac{34073}{21}a^{10}-\frac{53476}{21}a^{9}+6762a^{8}+\frac{35527}{42}a^{7}-\frac{620531}{42}a^{6}+\frac{392870}{21}a^{5}-\frac{506395}{42}a^{4}+\frac{28703}{6}a^{3}-\frac{24398}{21}a^{2}+\frac{3134}{21}a-\frac{109}{21}$, $\frac{1271}{21}a^{17}-\frac{635}{6}a^{16}-\frac{11299}{42}a^{15}+\frac{13840}{21}a^{14}-\frac{6259}{6}a^{13}+\frac{10009}{7}a^{12}+\frac{228805}{42}a^{11}-\frac{88743}{7}a^{10}-\frac{592757}{42}a^{9}+\frac{2140877}{42}a^{8}-\frac{153313}{21}a^{7}-\frac{615107}{6}a^{6}+\frac{2176845}{14}a^{5}-\frac{1628595}{14}a^{4}+\frac{322259}{6}a^{3}-\frac{223051}{14}a^{2}+\frac{118745}{42}a-\frac{3193}{14}$, $\frac{361}{6}a^{17}-\frac{2062}{21}a^{16}-\frac{1943}{7}a^{15}+\frac{4343}{7}a^{14}-\frac{6798}{7}a^{13}+\frac{3968}{3}a^{12}+\frac{77701}{14}a^{11}-\frac{500855}{42}a^{10}-\frac{107067}{7}a^{9}+\frac{2041457}{42}a^{8}-\frac{41098}{21}a^{7}-\frac{2120921}{21}a^{6}+142830a^{5}-\frac{709798}{7}a^{4}+\frac{625637}{14}a^{3}-\frac{264532}{21}a^{2}+\frac{43847}{21}a-\frac{2159}{14}$, $a-1$, $a^{17}-a^{16}-5a^{15}+7a^{14}-13a^{13}+15a^{12}+99a^{11}-133a^{10}-313a^{9}+587a^{8}+253a^{7}-1407a^{6}+1589a^{5}-987a^{4}+393a^{3}-99a^{2}+16a-1$, $\frac{341}{14}a^{17}-\frac{275}{6}a^{16}-\frac{4369}{42}a^{15}+\frac{3937}{14}a^{14}-\frac{2689}{6}a^{13}+\frac{13024}{21}a^{12}+\frac{44840}{21}a^{11}-\frac{227461}{42}a^{10}-\frac{107914}{21}a^{9}+\frac{451054}{21}a^{8}-\frac{110095}{21}a^{7}-41784a^{6}+\frac{474512}{7}a^{5}-\frac{742017}{14}a^{4}+\frac{76043}{3}a^{3}-\frac{326909}{42}a^{2}+\frac{30200}{21}a-\frac{5071}{42}$, $\frac{341}{14}a^{17}-\frac{275}{6}a^{16}-\frac{4369}{42}a^{15}+\frac{3937}{14}a^{14}-\frac{2689}{6}a^{13}+\frac{13024}{21}a^{12}+\frac{44840}{21}a^{11}-\frac{227461}{42}a^{10}-\frac{107914}{21}a^{9}+\frac{451054}{21}a^{8}-\frac{110095}{21}a^{7}-41784a^{6}+\frac{474512}{7}a^{5}-\frac{742017}{14}a^{4}+\frac{76043}{3}a^{3}-\frac{326909}{42}a^{2}+\frac{30200}{21}a-\frac{5113}{42}$, $\frac{1657}{14}a^{17}-\frac{8167}{42}a^{16}-\frac{3803}{7}a^{15}+\frac{17173}{14}a^{14}-\frac{80933}{42}a^{13}+\frac{55088}{21}a^{12}+\frac{228556}{21}a^{11}-\frac{330089}{14}a^{10}-\frac{416719}{14}a^{9}+\frac{2014811}{21}a^{8}-\frac{71829}{14}a^{7}-\frac{8338571}{42}a^{6}+\frac{5954584}{21}a^{5}-\frac{2842677}{14}a^{4}+\frac{1895596}{21}a^{3}-\frac{154291}{6}a^{2}+\frac{182291}{42}a-\frac{6899}{21}$, $\frac{166}{3}a^{17}-\frac{1327}{14}a^{16}-\frac{5210}{21}a^{15}+\frac{12448}{21}a^{14}-\frac{19715}{21}a^{13}+\frac{26902}{21}a^{12}+\frac{70157}{14}a^{11}-\frac{239345}{21}a^{10}-\frac{553793}{42}a^{9}+\frac{1931879}{42}a^{8}-\frac{228425}{42}a^{7}-\frac{652992}{7}a^{6}+\frac{2924261}{21}a^{5}-\frac{2163598}{21}a^{4}+\frac{1976939}{42}a^{3}-\frac{578695}{42}a^{2}+\frac{4807}{2}a-\frac{7999}{42}$, $\frac{1879}{21}a^{17}-\frac{1070}{7}a^{16}-\frac{5623}{14}a^{15}+\frac{20078}{21}a^{14}-\frac{63625}{42}a^{13}+\frac{43423}{21}a^{12}+\frac{340577}{42}a^{11}-\frac{772181}{42}a^{10}-\frac{898103}{42}a^{9}+\frac{222646}{3}a^{8}-\frac{357671}{42}a^{7}-\frac{2108853}{14}a^{6}+\frac{3142289}{14}a^{5}-\frac{6974741}{42}a^{4}+\frac{456283}{6}a^{3}-\frac{470104}{21}a^{2}+\frac{82865}{21}a-\frac{13367}{42}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 299765.553553 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 299765.553553 \cdot 1}{2\cdot\sqrt{4581441688047722385682249}}\cr\approx \mathstrut & 0.275746388337 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 4*x^16 + 12*x^15 - 20*x^14 + 28*x^13 + 84*x^12 - 232*x^11 - 180*x^10 + 900*x^9 - 334*x^8 - 1660*x^7 + 2996*x^6 - 2576*x^5 + 1380*x^4 - 492*x^3 + 115*x^2 - 16*x + 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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^18 - 2*x^17 - 4*x^16 + 12*x^15 - 20*x^14 + 28*x^13 + 84*x^12 - 232*x^11 - 180*x^10 + 900*x^9 - 334*x^8 - 1660*x^7 + 2996*x^6 - 2576*x^5 + 1380*x^4 - 492*x^3 + 115*x^2 - 16*x + 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 2*x^17 - 4*x^16 + 12*x^15 - 20*x^14 + 28*x^13 + 84*x^12 - 232*x^11 - 180*x^10 + 900*x^9 - 334*x^8 - 1660*x^7 + 2996*x^6 - 2576*x^5 + 1380*x^4 - 492*x^3 + 115*x^2 - 16*x + 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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 - 4*x^16 + 12*x^15 - 20*x^14 + 28*x^13 + 84*x^12 - 232*x^11 - 180*x^10 + 900*x^9 - 334*x^8 - 1660*x^7 + 2996*x^6 - 2576*x^5 + 1380*x^4 - 492*x^3 + 115*x^2 - 16*x + 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

$C_3^2.A_4$ (as 18T47):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_3^2.A_4$
Character table for $C_3^2.A_4$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.405769.1, 9.9.164648481361.1

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

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

Sibling fields

Degree 18 siblings: data not computed
Degree 36 sibling: 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 ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ R ${\href{/padicField/11.9.0.1}{9} }^{2}$ R ${\href{/padicField/17.9.0.1}{9} }^{2}$ ${\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.9.0.1}{9} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
\(7\) Copy content Toggle raw display 7.9.8.2$x^{9} + 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
7.9.8.2$x^{9} + 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
\(13\) Copy content Toggle raw display 13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.5.1$x^{6} + 52$$6$$1$$5$$C_6$$[\ ]_{6}$