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Properties

Label 18.0.581...247.1
Degree $18$
Signature $[0, 9]$
Discriminant $-5.812\times 10^{31}$
Root discriminant \(58.17\)
Ramified primes $13,19$
Class number $9774$ (GRH)
Class group [9774] (GRH)
Galois group $C_{18}$ (as 18T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353)
 
gp: K = bnfinit(y^18 - y^17 + 58*y^16 - 58*y^15 + 1426*y^14 - 1426*y^13 + 19381*y^12 - 19381*y^11 + 159430*y^10 - 159430*y^9 + 819661*y^8 - 819661*y^7 + 2647993*y^6 - 2647993*y^5 + 5390491*y^4 - 5390491*y^3 + 7260376*y^2 - 7260376*y + 7634353, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353)
 

\( x^{18} - x^{17} + 58 x^{16} - 58 x^{15} + 1426 x^{14} - 1426 x^{13} + 19381 x^{12} - 19381 x^{11} + \cdots + 7634353 \) 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:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-58116758997176374949719154916247\) \(\medspace = -\,13^{9}\cdot 19^{17}\) 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:  \(58.17\)
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:  $13^{1/2}19^{17/18}\approx 58.16790339829716$
Ramified primes:   \(13\), \(19\) 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:  \(\Q(\sqrt{-247}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(66,·)$, $\chi_{247}(131,·)$, $\chi_{247}(196,·)$, $\chi_{247}(118,·)$, $\chi_{247}(129,·)$, $\chi_{247}(12,·)$, $\chi_{247}(144,·)$, $\chi_{247}(90,·)$, $\chi_{247}(155,·)$, $\chi_{247}(92,·)$, $\chi_{247}(157,·)$, $\chi_{247}(103,·)$, $\chi_{247}(235,·)$, $\chi_{247}(51,·)$, $\chi_{247}(116,·)$, $\chi_{247}(181,·)$, $\chi_{247}(246,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{256}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2117473}a^{10}+\frac{562712}{2117473}a^{9}+\frac{30}{2117473}a^{8}+\frac{370913}{2117473}a^{7}+\frac{315}{2117473}a^{6}-\frac{896729}{2117473}a^{5}+\frac{1350}{2117473}a^{4}+\frac{540025}{2117473}a^{3}+\frac{2025}{2117473}a^{2}-\frac{572714}{2117473}a+\frac{486}{2117473}$, $\frac{1}{2117473}a^{11}+\frac{33}{2117473}a^{9}+\frac{429337}{2117473}a^{8}+\frac{396}{2117473}a^{7}-\frac{283277}{2117473}a^{6}+\frac{2079}{2117473}a^{5}+\frac{1051632}{2117473}a^{4}+\frac{4455}{2117473}a^{3}-\frac{864040}{2117473}a^{2}+\frac{2673}{2117473}a-\frac{324015}{2117473}$, $\frac{1}{2117473}a^{12}+\frac{917098}{2117473}a^{9}-\frac{594}{2117473}a^{8}+\frac{181432}{2117473}a^{7}-\frac{8316}{2117473}a^{6}+\frac{999067}{2117473}a^{5}-\frac{40095}{2117473}a^{4}+\frac{372392}{2117473}a^{3}-\frac{64152}{2117473}a^{2}-\frac{481710}{2117473}a-\frac{16038}{2117473}$, $\frac{1}{2117473}a^{13}-\frac{702}{2117473}a^{9}+\frac{195641}{2117473}a^{8}-\frac{11232}{2117473}a^{7}+\frac{89525}{2117473}a^{6}-\frac{66339}{2117473}a^{5}+\frac{1011797}{2117473}a^{4}-\frac{151632}{2117473}a^{3}-\frac{581339}{2117473}a^{2}-\frac{94770}{2117473}a-\frac{1040298}{2117473}$, $\frac{1}{2117473}a^{14}-\frac{747986}{2117473}a^{9}+\frac{9828}{2117473}a^{8}+\frac{21272}{2117473}a^{7}+\frac{154791}{2117473}a^{6}+\frac{397520}{2117473}a^{5}+\frac{796068}{2117473}a^{4}-\frac{511456}{2117473}a^{3}-\frac{790693}{2117473}a^{2}-\frac{765656}{2117473}a+\frac{341172}{2117473}$, $\frac{1}{2117473}a^{15}+\frac{12285}{2117473}a^{9}-\frac{831351}{2117473}a^{8}+\frac{221130}{2117473}a^{7}+\frac{973607}{2117473}a^{6}-\frac{724354}{2117473}a^{5}-\frac{764977}{2117473}a^{4}-\frac{917996}{2117473}a^{3}-\frac{87201}{2117473}a^{2}+\frac{14852}{2117473}a-\frac{684160}{2117473}$, $\frac{1}{2117473}a^{16}-\frac{198926}{2117473}a^{9}-\frac{147420}{2117473}a^{8}-\frac{1008175}{2117473}a^{7}-\frac{359183}{2117473}a^{6}+\frac{456242}{2117473}a^{5}-\frac{562962}{2117473}a^{4}-\frac{251417}{2117473}a^{3}+\frac{547403}{2117473}a^{2}+\frac{862024}{2117473}a+\frac{381909}{2117473}$, $\frac{1}{2117473}a^{17}-\frac{192780}{2117473}a^{9}+\frac{724659}{2117473}a^{8}+\frac{533570}{2117473}a^{7}-\frac{406258}{2117473}a^{6}-\frac{998077}{2117473}a^{5}-\frac{620388}{2117473}a^{4}-\frac{197156}{2117473}a^{3}-\frac{750169}{2117473}a^{2}-\frac{923436}{2117473}a-\frac{725722}{2117473}$ 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

$C_{9774}$, which has order $9774$ (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:  $8$
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:   $\frac{97}{2117473}a^{12}+\frac{3492}{2117473}a^{10}+\frac{47142}{2117473}a^{8}-\frac{6160}{2117473}a^{7}+\frac{293328}{2117473}a^{6}-\frac{129360}{2117473}a^{5}+\frac{824985}{2117473}a^{4}-\frac{776160}{2117473}a^{3}+\frac{848556}{2117473}a^{2}-\frac{1164240}{2117473}a+\frac{141426}{2117473}$, $\frac{217}{2117473}a^{11}+\frac{7161}{2117473}a^{9}-\frac{2683}{2117473}a^{8}+\frac{85932}{2117473}a^{7}-\frac{64392}{2117473}a^{6}+\frac{451143}{2117473}a^{5}-\frac{482940}{2117473}a^{4}+\frac{966735}{2117473}a^{3}-\frac{1159056}{2117473}a^{2}+\frac{580041}{2117473}a-\frac{434646}{2117473}$, $\frac{4}{2117473}a^{16}+\frac{192}{2117473}a^{14}+\frac{3744}{2117473}a^{12}+\frac{38016}{2117473}a^{10}+\frac{213840}{2117473}a^{8}+\frac{653184}{2117473}a^{6}+\frac{979776}{2117473}a^{4}-\frac{173383}{2117473}a^{3}+\frac{559872}{2117473}a^{2}-\frac{1560447}{2117473}a+\frac{2169961}{2117473}$, $\frac{40}{2117473}a^{13}+\frac{1560}{2117473}a^{11}+\frac{23400}{2117473}a^{9}+\frac{168480}{2117473}a^{7}-\frac{14209}{2117473}a^{6}+\frac{589680}{2117473}a^{5}-\frac{255762}{2117473}a^{4}+\frac{884520}{2117473}a^{3}-\frac{1150929}{2117473}a^{2}+\frac{379080}{2117473}a-\frac{767286}{2117473}$, $\frac{19}{2117473}a^{14}+\frac{798}{2117473}a^{12}+\frac{13167}{2117473}a^{10}+\frac{107730}{2117473}a^{8}+\frac{452466}{2117473}a^{6}-\frac{32689}{2117473}a^{5}+\frac{904932}{2117473}a^{4}-\frac{490335}{2117473}a^{3}+\frac{678699}{2117473}a^{2}-\frac{1471005}{2117473}a+\frac{83106}{2117473}$, $\frac{1}{2117473}a^{17}+\frac{51}{2117473}a^{15}+\frac{1071}{2117473}a^{13}+\frac{11934}{2117473}a^{11}+\frac{75735}{2117473}a^{9}+\frac{272646}{2117473}a^{7}+\frac{520506}{2117473}a^{5}+\frac{446148}{2117473}a^{3}-\frac{399331}{2117473}a^{2}+\frac{111537}{2117473}a-\frac{4513459}{2117473}$, $\frac{19}{2117473}a^{14}+\frac{798}{2117473}a^{12}+\frac{13167}{2117473}a^{10}+\frac{107730}{2117473}a^{8}+\frac{452466}{2117473}a^{6}-\frac{32689}{2117473}a^{5}+\frac{904932}{2117473}a^{4}-\frac{490335}{2117473}a^{3}+\frac{678699}{2117473}a^{2}-\frac{1471005}{2117473}a+\frac{2200579}{2117473}$, $\frac{7}{2117473}a^{15}+\frac{315}{2117473}a^{13}+\frac{5670}{2117473}a^{11}+\frac{51975}{2117473}a^{9}+\frac{255150}{2117473}a^{7}+\frac{642978}{2117473}a^{5}-\frac{75316}{2117473}a^{4}+\frac{714420}{2117473}a^{3}-\frac{903792}{2117473}a^{2}+\frac{229635}{2117473}a-\frac{1355688}{2117473}$ 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:  \( 22305.8950792 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22305.8950792 \cdot 9774}{2\cdot\sqrt{58116758997176374949719154916247}}\cr\approx \mathstrut & 0.218237870780 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353)
 
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 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353, 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 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353);
 
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 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353);
 
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

$C_{18}$ (as 18T1):

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 cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-247}) \), 3.3.361.1, 6.0.5439989503.1, \(\Q(\zeta_{19})^+\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ R ${\href{/padicField/17.9.0.1}{9} }^{2}$ R ${\href{/padicField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.1.0.1}{1} }^{18}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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
\(13\) Copy content Toggle raw display 13.18.9.2$x^{18} - 4455516 x^{8} + 38614472 x^{6} - 752982204 x^{4} + 9788768652 x^{2} - 116649493103$$2$$9$$9$$C_{18}$$[\ ]_{2}^{9}$
\(19\) Copy content Toggle raw display 19.18.17.1$x^{18} + 342$$18$$1$$17$$C_{18}$$[\ ]_{18}$