↑

Properties

Label 18.6.160...000.1
Degree $18$
Signature $[6, 6]$
Discriminant $1.606\times 10^{60}$
Root discriminant \(2211.91\)
Ramified primes $2,3,5,17,31$
Class number not computed
Class group not computed
Galois group $C_3^2:D_6$ (as 18T52)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 840*x^16 - 39148*x^15 + 372102*x^14 + 34410474*x^13 + 438897946*x^12 - 6292646688*x^11 - 565840514061*x^10 - 2002216086899*x^9 + 61678996944750*x^8 + 4303504696552764*x^7 + 26352158241463094*x^6 - 1056477032125368300*x^5 - 5760620986430140224*x^4 - 290620036033694106616*x^3 + 5882949351624155941488*x^2 + 425577526429058792736*x - 24688926265565708628544)
 
gp: K = bnfinit(y^18 - 3*y^17 - 840*y^16 - 39148*y^15 + 372102*y^14 + 34410474*y^13 + 438897946*y^12 - 6292646688*y^11 - 565840514061*y^10 - 2002216086899*y^9 + 61678996944750*y^8 + 4303504696552764*y^7 + 26352158241463094*y^6 - 1056477032125368300*y^5 - 5760620986430140224*y^4 - 290620036033694106616*y^3 + 5882949351624155941488*y^2 + 425577526429058792736*y - 24688926265565708628544, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 840*x^16 - 39148*x^15 + 372102*x^14 + 34410474*x^13 + 438897946*x^12 - 6292646688*x^11 - 565840514061*x^10 - 2002216086899*x^9 + 61678996944750*x^8 + 4303504696552764*x^7 + 26352158241463094*x^6 - 1056477032125368300*x^5 - 5760620986430140224*x^4 - 290620036033694106616*x^3 + 5882949351624155941488*x^2 + 425577526429058792736*x - 24688926265565708628544);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 - 840*x^16 - 39148*x^15 + 372102*x^14 + 34410474*x^13 + 438897946*x^12 - 6292646688*x^11 - 565840514061*x^10 - 2002216086899*x^9 + 61678996944750*x^8 + 4303504696552764*x^7 + 26352158241463094*x^6 - 1056477032125368300*x^5 - 5760620986430140224*x^4 - 290620036033694106616*x^3 + 5882949351624155941488*x^2 + 425577526429058792736*x - 24688926265565708628544)
 

\( x^{18} - 3 x^{17} - 840 x^{16} - 39148 x^{15} + 372102 x^{14} + 34410474 x^{13} + 438897946 x^{12} + \cdots - 24\!\cdots\!44 \) 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:   \(1606252976737815398749110318939665274672826713993000000000000\) \(\medspace = 2^{12}\cdot 3^{30}\cdot 5^{12}\cdot 17^{13}\cdot 31^{12}\) 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:  \(2211.91\)
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:  $2^{2/3}3^{11/6}5^{2/3}17^{5/6}31^{2/3}\approx 3639.1855330031426$
Ramified primes:   \(2\), \(3\), \(5\), \(17\), \(31\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{9}+\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{2}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{36}a^{10}+\frac{1}{36}a^{9}+\frac{1}{18}a^{8}+\frac{1}{18}a^{7}+\frac{1}{36}a^{6}-\frac{1}{12}a^{5}-\frac{1}{18}a^{4}-\frac{2}{9}a^{3}+\frac{1}{18}a^{2}-\frac{1}{9}a+\frac{4}{9}$, $\frac{1}{36}a^{11}+\frac{1}{36}a^{9}-\frac{1}{36}a^{7}-\frac{1}{9}a^{6}+\frac{1}{36}a^{5}-\frac{1}{6}a^{4}+\frac{5}{18}a^{3}-\frac{1}{6}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{1224}a^{12}+\frac{1}{612}a^{11}+\frac{7}{1224}a^{10}+\frac{23}{306}a^{9}+\frac{35}{1224}a^{8}-\frac{25}{204}a^{7}+\frac{107}{1224}a^{6}-\frac{7}{36}a^{5}+\frac{227}{612}a^{4}-\frac{281}{612}a^{3}+\frac{11}{306}a^{2}+\frac{25}{51}a+\frac{44}{153}$, $\frac{1}{1224}a^{13}+\frac{1}{408}a^{11}+\frac{5}{612}a^{10}-\frac{13}{1224}a^{9}+\frac{13}{306}a^{8}+\frac{67}{1224}a^{7}+\frac{23}{306}a^{6}-\frac{5}{68}a^{5}+\frac{251}{612}a^{4}+\frac{10}{153}a^{3}-\frac{55}{153}a^{2}-\frac{7}{51}a-\frac{71}{153}$, $\frac{1}{19584}a^{14}+\frac{7}{19584}a^{13}-\frac{7}{19584}a^{12}-\frac{19}{6528}a^{11}-\frac{9}{2176}a^{10}+\frac{383}{6528}a^{9}-\frac{463}{19584}a^{8}-\frac{863}{19584}a^{7}-\frac{115}{816}a^{6}+\frac{11}{51}a^{5}-\frac{67}{3264}a^{4}-\frac{207}{544}a^{3}+\frac{899}{2448}a^{2}-\frac{79}{816}a-\frac{545}{1224}$, $\frac{1}{19584}a^{15}-\frac{1}{2448}a^{13}-\frac{1}{2448}a^{12}-\frac{41}{9792}a^{11}+\frac{5}{4896}a^{10}+\frac{67}{1088}a^{9}+\frac{29}{1088}a^{8}-\frac{3839}{19584}a^{7}-\frac{65}{272}a^{6}-\frac{4361}{9792}a^{5}+\frac{3745}{9792}a^{4}-\frac{2425}{4896}a^{3}+\frac{27}{136}a^{2}+\frac{307}{816}a-\frac{337}{1224}$, $\frac{1}{24\!\cdots\!64}a^{16}-\frac{32\!\cdots\!35}{13\!\cdots\!48}a^{15}-\frac{14\!\cdots\!27}{67\!\cdots\!24}a^{14}-\frac{65\!\cdots\!77}{60\!\cdots\!16}a^{13}-\frac{20\!\cdots\!63}{12\!\cdots\!32}a^{12}-\frac{70\!\cdots\!03}{60\!\cdots\!16}a^{11}+\frac{11\!\cdots\!89}{12\!\cdots\!32}a^{10}-\frac{27\!\cdots\!63}{12\!\cdots\!32}a^{9}+\frac{62\!\cdots\!53}{14\!\cdots\!92}a^{8}+\frac{84\!\cdots\!89}{40\!\cdots\!44}a^{7}-\frac{19\!\cdots\!93}{13\!\cdots\!48}a^{6}+\frac{28\!\cdots\!63}{71\!\cdots\!96}a^{5}-\frac{14\!\cdots\!17}{30\!\cdots\!08}a^{4}-\frac{12\!\cdots\!53}{30\!\cdots\!08}a^{3}-\frac{48\!\cdots\!05}{13\!\cdots\!96}a^{2}+\frac{91\!\cdots\!55}{75\!\cdots\!52}a+\frac{34\!\cdots\!29}{75\!\cdots\!52}$, $\frac{1}{24\!\cdots\!64}a^{17}+\frac{20\!\cdots\!57}{12\!\cdots\!32}a^{16}+\frac{49\!\cdots\!61}{61\!\cdots\!16}a^{15}+\frac{49\!\cdots\!63}{61\!\cdots\!16}a^{14}+\frac{11\!\cdots\!57}{12\!\cdots\!32}a^{13}-\frac{58\!\cdots\!11}{61\!\cdots\!16}a^{12}-\frac{75\!\cdots\!03}{12\!\cdots\!32}a^{11}-\frac{32\!\cdots\!13}{24\!\cdots\!32}a^{10}+\frac{45\!\cdots\!07}{82\!\cdots\!88}a^{9}+\frac{11\!\cdots\!87}{12\!\cdots\!32}a^{8}+\frac{11\!\cdots\!83}{12\!\cdots\!32}a^{7}-\frac{24\!\cdots\!81}{12\!\cdots\!32}a^{6}-\frac{10\!\cdots\!23}{30\!\cdots\!08}a^{5}+\frac{43\!\cdots\!59}{91\!\cdots\!84}a^{4}+\frac{17\!\cdots\!61}{30\!\cdots\!08}a^{3}+\frac{85\!\cdots\!21}{77\!\cdots\!52}a^{2}-\frac{50\!\cdots\!95}{77\!\cdots\!52}a+\frac{30\!\cdots\!29}{12\!\cdots\!43}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

not computed

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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot R \cdot h}{2\cdot\sqrt{1606252976737815398749110318939665274672826713993000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 840*x^16 - 39148*x^15 + 372102*x^14 + 34410474*x^13 + 438897946*x^12 - 6292646688*x^11 - 565840514061*x^10 - 2002216086899*x^9 + 61678996944750*x^8 + 4303504696552764*x^7 + 26352158241463094*x^6 - 1056477032125368300*x^5 - 5760620986430140224*x^4 - 290620036033694106616*x^3 + 5882949351624155941488*x^2 + 425577526429058792736*x - 24688926265565708628544)
 
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 - 3*x^17 - 840*x^16 - 39148*x^15 + 372102*x^14 + 34410474*x^13 + 438897946*x^12 - 6292646688*x^11 - 565840514061*x^10 - 2002216086899*x^9 + 61678996944750*x^8 + 4303504696552764*x^7 + 26352158241463094*x^6 - 1056477032125368300*x^5 - 5760620986430140224*x^4 - 290620036033694106616*x^3 + 5882949351624155941488*x^2 + 425577526429058792736*x - 24688926265565708628544, 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 - 3*x^17 - 840*x^16 - 39148*x^15 + 372102*x^14 + 34410474*x^13 + 438897946*x^12 - 6292646688*x^11 - 565840514061*x^10 - 2002216086899*x^9 + 61678996944750*x^8 + 4303504696552764*x^7 + 26352158241463094*x^6 - 1056477032125368300*x^5 - 5760620986430140224*x^4 - 290620036033694106616*x^3 + 5882949351624155941488*x^2 + 425577526429058792736*x - 24688926265565708628544);
 
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 - 3*x^17 - 840*x^16 - 39148*x^15 + 372102*x^14 + 34410474*x^13 + 438897946*x^12 - 6292646688*x^11 - 565840514061*x^10 - 2002216086899*x^9 + 61678996944750*x^8 + 4303504696552764*x^7 + 26352158241463094*x^6 - 1056477032125368300*x^5 - 5760620986430140224*x^4 - 290620036033694106616*x^3 + 5882949351624155941488*x^2 + 425577526429058792736*x - 24688926265565708628544);
 
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

$C_3^2:D_6$ (as 18T52):

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 $C_3^2:D_6$
Character table for $C_3^2:D_6$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.1.24300.2, 6.2.2901077370000.16, 9.3.1063615528562424054507000000.2

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 siblings: data not computed
Minimal sibling: 18.6.7075249356458293708862869717627850394458877005700000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ R ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ R ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$

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
\(2\) Copy content Toggle raw display 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.6.10.2$x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.2$x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.2$x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(17\) Copy content Toggle raw display 17.6.3.1$x^{6} + 459 x^{5} + 70280 x^{4} + 3597823 x^{3} + 1271380 x^{2} + 4696159 x + 50437479$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.12.10.2$x^{12} - 3060 x^{6} - 197676$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$
\(31\) Copy content Toggle raw display 31.6.4.1$x^{6} + 87 x^{5} + 2532 x^{4} + 24973 x^{3} + 10293 x^{2} + 78438 x + 748956$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.6.4.2$x^{6} - 899 x^{3} + 2883$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.6.4.3$x^{6} - 62 x^{3} - 795708$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$