Properties

Label 18.2.808...433.1
Degree $18$
Signature $[2, 8]$
Discriminant $8.084\times 10^{29}$
Root discriminant \(45.87\)
Ramified primes $3,17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_9$ (as 18T28)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166)
 
gp: K = bnfinit(y^18 - 9*y^17 + 36*y^16 - 84*y^15 + 117*y^14 - 63*y^13 - 126*y^12 + 405*y^11 - 549*y^10 + 248*y^9 + 441*y^8 - 981*y^7 + 834*y^6 - 171*y^5 - 783*y^4 + 1269*y^3 - 36*y^2 - 549*y + 166, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166)
 

\( x^{18} - 9 x^{17} + 36 x^{16} - 84 x^{15} + 117 x^{14} - 63 x^{13} - 126 x^{12} + 405 x^{11} - 549 x^{10} + 248 x^{9} + 441 x^{8} - 981 x^{7} + 834 x^{6} - 171 x^{5} - 783 x^{4} + \cdots + 166 \) 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:  $[2, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(808432815650446861638683444433\) \(\medspace = 3^{24}\cdot 17^{15}\) 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:  \(45.87\)
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:  $3^{25/18}17^{7/8}\approx 54.86700011448388$
Ramified primes:   \(3\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{4}$, $\frac{1}{12}a^{14}+\frac{1}{12}a^{13}-\frac{1}{12}a^{11}-\frac{1}{12}a^{10}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{12}a^{5}-\frac{5}{12}a^{4}+\frac{5}{12}a^{2}+\frac{5}{12}a-\frac{1}{2}$, $\frac{1}{12}a^{15}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}+\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}+\frac{1}{12}a^{6}-\frac{1}{2}a^{5}+\frac{5}{12}a^{4}+\frac{5}{12}a^{3}+\frac{1}{12}a+\frac{1}{6}$, $\frac{1}{34116}a^{16}-\frac{2}{8529}a^{15}+\frac{61}{8529}a^{14}-\frac{392}{8529}a^{13}-\frac{691}{8529}a^{12}+\frac{622}{8529}a^{11}-\frac{553}{5686}a^{10}-\frac{236}{2843}a^{9}+\frac{408}{2843}a^{8}-\frac{3806}{8529}a^{7}-\frac{3157}{17058}a^{6}-\frac{6007}{17058}a^{5}-\frac{2591}{8529}a^{4}+\frac{368}{8529}a^{3}-\frac{247}{17058}a^{2}-\frac{3687}{11372}a+\frac{1689}{5686}$, $\frac{1}{3445716}a^{17}+\frac{7}{574286}a^{16}-\frac{2869}{574286}a^{15}+\frac{46189}{1148572}a^{14}+\frac{325385}{3445716}a^{13}+\frac{80677}{574286}a^{12}+\frac{470771}{3445716}a^{11}-\frac{404701}{3445716}a^{10}-\frac{258833}{1722858}a^{9}-\frac{1058303}{3445716}a^{8}+\frac{36991}{1148572}a^{7}-\frac{51475}{287143}a^{6}-\frac{411227}{1148572}a^{5}+\frac{930359}{3445716}a^{4}-\frac{115683}{287143}a^{3}+\frac{148435}{1722858}a^{2}-\frac{1410031}{3445716}a+\frac{79927}{1722858}$ 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

Trivial group, which has order $1$ (assuming GRH)

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:  $9$
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:   $\frac{98949}{1148572}a^{17}-\frac{2489213}{3445716}a^{16}+\frac{3058937}{1148572}a^{15}-\frac{19430723}{3445716}a^{14}+\frac{11603843}{1722858}a^{13}-\frac{1731997}{1148572}a^{12}-\frac{39898051}{3445716}a^{11}+\frac{7993649}{287143}a^{10}-\frac{106098707}{3445716}a^{9}+\frac{4152885}{1148572}a^{8}+\frac{33769096}{861429}a^{7}-\frac{69558691}{1148572}a^{6}+\frac{125602357}{3445716}a^{5}+\frac{8972729}{1722858}a^{4}-\frac{71695157}{1148572}a^{3}+\frac{61042139}{861429}a^{2}+\frac{42548249}{1148572}a-\frac{40262251}{1722858}$, $\frac{48455}{3445716}a^{17}-\frac{106262}{861429}a^{16}+\frac{409325}{861429}a^{15}-\frac{1798091}{1722858}a^{14}+\frac{739693}{574286}a^{13}-\frac{289247}{861429}a^{12}-\frac{609741}{287143}a^{11}+\frac{8887925}{1722858}a^{10}-\frac{5003107}{861429}a^{9}+\frac{1473751}{1722858}a^{8}+\frac{6343417}{861429}a^{7}-\frac{19745801}{1722858}a^{6}+\frac{11692345}{1722858}a^{5}+\frac{746291}{574286}a^{4}-\frac{19224397}{1722858}a^{3}+\frac{14918721}{1148572}a^{2}+\frac{4851499}{861429}a-\frac{4883194}{861429}$, $\frac{1989}{287143}a^{17}-\frac{30671}{861429}a^{16}+\frac{11464}{287143}a^{15}+\frac{138229}{861429}a^{14}-\frac{659725}{861429}a^{13}+\frac{1350728}{861429}a^{12}-\frac{511572}{287143}a^{11}+\frac{397786}{861429}a^{10}+\frac{2820097}{861429}a^{9}-\frac{2131941}{287143}a^{8}+\frac{6747703}{861429}a^{7}-\frac{668106}{287143}a^{6}-\frac{4798844}{861429}a^{5}+\frac{7744256}{861429}a^{4}-\frac{9905587}{861429}a^{3}-\frac{473338}{287143}a^{2}+\frac{6912478}{861429}a-\frac{1262429}{861429}$, $\frac{22625}{3445716}a^{17}-\frac{18631}{287143}a^{16}+\frac{962587}{3445716}a^{15}-\frac{594260}{861429}a^{14}+\frac{3475373}{3445716}a^{13}-\frac{2013163}{3445716}a^{12}-\frac{578661}{574286}a^{11}+\frac{11475323}{3445716}a^{10}-\frac{15700895}{3445716}a^{9}+\frac{1656722}{861429}a^{8}+\frac{4467659}{1148572}a^{7}-\frac{28862855}{3445716}a^{6}+\frac{10828025}{1722858}a^{5}-\frac{690085}{3445716}a^{4}-\frac{19311151}{3445716}a^{3}+\frac{14344165}{1148572}a^{2}+\frac{11883185}{3445716}a-\frac{7994215}{1722858}$, $\frac{100595}{1148572}a^{17}-\frac{2118803}{3445716}a^{16}+\frac{535715}{287143}a^{15}-\frac{1780779}{574286}a^{14}+\frac{1162983}{574286}a^{13}+\frac{913376}{287143}a^{12}-\frac{9736559}{861429}a^{11}+\frac{4995206}{287143}a^{10}-\frac{7649381}{861429}a^{9}-\frac{8697855}{574286}a^{8}+\frac{31442077}{861429}a^{7}-\frac{8969511}{287143}a^{6}+\frac{7920}{287143}a^{5}+\frac{12832329}{574286}a^{4}-\frac{37290285}{574286}a^{3}+\frac{25315759}{3445716}a^{2}+\frac{40274629}{1148572}a-\frac{20422813}{1722858}$, $\frac{326483}{3445716}a^{17}-\frac{2814647}{3445716}a^{16}+\frac{10638409}{3445716}a^{15}-\frac{23090875}{3445716}a^{14}+\frac{7168685}{861429}a^{13}-\frac{8640313}{3445716}a^{12}-\frac{44656943}{3445716}a^{11}+\frac{56609239}{1722858}a^{10}-\frac{131483689}{3445716}a^{9}+\frac{25238423}{3445716}a^{8}+\frac{76567841}{1722858}a^{7}-\frac{252535847}{3445716}a^{6}+\frac{160769393}{3445716}a^{5}+\frac{3649742}{861429}a^{4}-\frac{243429721}{3445716}a^{3}+\frac{78069532}{861429}a^{2}+\frac{136903091}{3445716}a-\frac{57475547}{1722858}$, $\frac{18659}{3445716}a^{17}-\frac{146431}{3445716}a^{16}+\frac{666431}{3445716}a^{15}-\frac{1013773}{1722858}a^{14}+\frac{1441961}{1148572}a^{13}-\frac{6544499}{3445716}a^{12}+\frac{1532206}{861429}a^{11}-\frac{238311}{1148572}a^{10}-\frac{10444859}{3445716}a^{9}+\frac{10879237}{1722858}a^{8}-\frac{22817749}{3445716}a^{7}+\frac{9216743}{3445716}a^{6}+\frac{6610109}{1722858}a^{5}-\frac{8510777}{1148572}a^{4}+\frac{29799685}{3445716}a^{3}-\frac{1656095}{3445716}a^{2}-\frac{3225565}{574286}a+\frac{1842883}{861429}$, $\frac{307171}{3445716}a^{17}-\frac{1329641}{1722858}a^{16}+\frac{2646173}{861429}a^{15}-\frac{6430354}{861429}a^{14}+\frac{10213600}{861429}a^{13}-\frac{9627502}{861429}a^{12}-\frac{59137}{1722858}a^{11}+\frac{6436240}{287143}a^{10}-\frac{36387931}{861429}a^{9}+\frac{32073928}{861429}a^{8}-\frac{2779607}{1722858}a^{7}-\frac{71551493}{1722858}a^{6}+\frac{53559848}{861429}a^{5}-\frac{41374088}{861429}a^{4}-\frac{19556501}{1722858}a^{3}+\frac{156099157}{3445716}a^{2}-\frac{4160014}{287143}a-\frac{164494}{861429}$, $\frac{307171}{3445716}a^{17}-\frac{2562625}{3445716}a^{16}+\frac{2452859}{861429}a^{15}-\frac{5702851}{861429}a^{14}+\frac{8504074}{861429}a^{13}-\frac{2405342}{287143}a^{12}-\frac{2140747}{1722858}a^{11}+\frac{32172125}{1722858}a^{10}-\frac{28441049}{861429}a^{9}+\frac{23752033}{861429}a^{8}+\frac{2296249}{1722858}a^{7}-\frac{28123027}{861429}a^{6}+\frac{74369941}{1722858}a^{5}-\frac{31660817}{861429}a^{4}-\frac{3862803}{574286}a^{3}+\frac{115242031}{3445716}a^{2}-\frac{25602701}{3445716}a-\frac{5602501}{1722858}$ Copy content Toggle raw display (assuming GRH)
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:  \( 250225810.428 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 250225810.428 \cdot 1}{2\cdot\sqrt{808432815650446861638683444433}}\cr\approx \mathstrut & 1.35200743553 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166)
 
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 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166, 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 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166);
 
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 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166);
 
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

$F_9$ (as 18T28):

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 72
The 9 conjugacy class representatives for $F_9$
Character table for $F_9$

Intermediate fields

\(\Q(\sqrt{17}) \), 9.1.218070794717793.1

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 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 24 sibling: data not computed
Degree 36 sibling: data not computed
Minimal sibling: 9.1.218070794717793.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ R ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3\) Copy content Toggle raw display Deg $18$$9$$2$$24$
\(17\) Copy content Toggle raw display 17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.7.1$x^{8} + 68$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} + 68$$8$$1$$7$$C_8$$[\ ]_{8}$