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Properties

Label 18.0.925...000.4
Degree $18$
Signature $[0, 9]$
Discriminant $-9.255\times 10^{27}$
Root discriminant \(35.78\)
Ramified primes $2,3,5$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200)
 
gp: K = bnfinit(y^18 - 6*y^16 + 54*y^14 - 996*y^12 + 3321*y^10 + 5274*y^8 - 256*y^6 - 13920*y^4 - 38400*y^2 + 51200, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200)
 

\( x^{18} - 6x^{16} + 54x^{14} - 996x^{12} + 3321x^{10} + 5274x^{8} - 256x^{6} - 13920x^{4} - 38400x^{2} + 51200 \) 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:   \(-9254651051409408000000000000\) \(\medspace = -\,2^{27}\cdot 3^{24}\cdot 5^{12}\) 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:  \(35.78\)
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:  $2^{3/2}3^{4/3}5^{2/3}\approx 35.783817426546634$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q(\sqrt{-2}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}+\frac{1}{16}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{112}a^{12}-\frac{1}{16}a^{10}-\frac{1}{112}a^{8}+\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{3}{28}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{7}$, $\frac{1}{224}a^{13}-\frac{1}{224}a^{12}-\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{224}a^{9}-\frac{13}{224}a^{8}-\frac{3}{32}a^{7}+\frac{1}{32}a^{6}-\frac{5}{28}a^{5}+\frac{27}{112}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{14}a-\frac{3}{7}$, $\frac{1}{896}a^{14}+\frac{1}{448}a^{12}-\frac{1}{32}a^{11}+\frac{17}{448}a^{10}+\frac{1}{32}a^{9}-\frac{1}{56}a^{8}-\frac{1}{32}a^{7}+\frac{37}{896}a^{6}+\frac{7}{32}a^{5}-\frac{33}{448}a^{4}+\frac{5}{16}a^{3}+\frac{33}{112}a^{2}+\frac{3}{14}$, $\frac{1}{8960}a^{15}+\frac{1}{640}a^{13}-\frac{1}{224}a^{12}-\frac{53}{4480}a^{11}+\frac{1}{32}a^{10}+\frac{13}{320}a^{9}+\frac{1}{224}a^{8}-\frac{719}{8960}a^{7}+\frac{3}{32}a^{6}+\frac{61}{640}a^{5}-\frac{1}{14}a^{4}-\frac{107}{1120}a^{3}+\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{3}{7}$, $\frac{1}{3381903042560}a^{16}+\frac{129664951}{241564503040}a^{14}-\frac{287145611}{73519631360}a^{12}+\frac{5462833333}{120782251520}a^{10}+\frac{122025720281}{3381903042560}a^{8}-\frac{1}{8}a^{7}-\frac{325688303}{10502804480}a^{6}+\frac{3303534029}{211368940160}a^{4}-\frac{1}{8}a^{3}-\frac{1163019049}{3019556288}a^{2}-\frac{1}{4}a+\frac{826731057}{2642111752}$, $\frac{1}{13527612170240}a^{17}-\frac{32096993}{966258012160}a^{15}-\frac{63993375}{58815705088}a^{13}-\frac{1722091159}{96625801216}a^{11}-\frac{1}{16}a^{10}-\frac{732508709223}{13527612170240}a^{9}-\frac{1}{16}a^{8}-\frac{3802397911}{42011217920}a^{7}-\frac{1}{16}a^{6}-\frac{6907056141}{169095152128}a^{5}+\frac{1}{16}a^{4}+\frac{944802017}{8627303680}a^{3}-\frac{3}{8}a^{2}-\frac{3513881107}{10568447008}a-\frac{1}{2}$ 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

$C_{3}$, which has order $3$ (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:   $\frac{5136644601}{6763806085120}a^{17}+\frac{55835079}{73519631360}a^{16}-\frac{1696081193}{483129006080}a^{15}-\frac{19019151}{5251402240}a^{14}+\frac{1067847033}{29407852544}a^{13}+\frac{1333735733}{36759815680}a^{12}-\frac{34123920591}{48312900608}a^{11}-\frac{1870041693}{2625701120}a^{10}+\frac{10537771690577}{6763806085120}a^{9}+\frac{119667912879}{73519631360}a^{8}+\frac{127449071409}{21005608960}a^{7}+\frac{32332339169}{5251402240}a^{6}+\frac{685815063399}{84547576064}a^{5}+\frac{35208189711}{4594976960}a^{4}+\frac{8313630279}{30195562880}a^{3}+\frac{8003841}{65642528}a^{2}-\frac{157386843027}{5284223504}a-\frac{1665819881}{57437212}$, $\frac{88482729}{845475760640}a^{17}+\frac{37524237}{147039262720}a^{16}-\frac{1968977}{4313651840}a^{15}-\frac{13052133}{10502804480}a^{14}+\frac{88301679}{18379907840}a^{13}+\frac{909312199}{73519631360}a^{12}-\frac{1443312411}{15097781440}a^{11}-\frac{1265073279}{5251402240}a^{10}+\frac{156527796457}{845475760640}a^{9}+\frac{84457453317}{147039262720}a^{8}+\frac{1267676427}{1312850560}a^{7}+\frac{20576179307}{10502804480}a^{6}+\frac{231973051731}{211368940160}a^{5}+\frac{23667216933}{9189953920}a^{4}+\frac{852857449}{7548890720}a^{3}-\frac{4381923}{18755008}a^{2}-\frac{1585079313}{330263969}a-\frac{1120737587}{114874424}$, $\frac{56095969}{483129006080}a^{17}-\frac{162423}{5163210752}a^{16}-\frac{143472739}{241564503040}a^{15}+\frac{31503}{368800768}a^{14}+\frac{1720591}{300080128}a^{13}-\frac{145795}{112243712}a^{12}-\frac{2666666213}{24156450304}a^{11}+\frac{4913357}{184400384}a^{10}+\frac{138904503513}{483129006080}a^{9}-\frac{63734943}{5163210752}a^{8}+\frac{9270802747}{10502804480}a^{7}-\frac{5082727}{16034816}a^{6}+\frac{276688579}{431365184}a^{5}-\frac{295869955}{322700672}a^{4}-\frac{19104058893}{15097781440}a^{3}-\frac{4662475}{3292864}a^{2}-\frac{2286367687}{377444536}a-\frac{9257627}{20168792}$, $\frac{162423}{2581605376}a^{16}-\frac{31503}{184400384}a^{14}+\frac{145795}{56121856}a^{12}-\frac{4913357}{92200192}a^{10}+\frac{63734943}{2581605376}a^{8}+\frac{5082727}{8017408}a^{6}+\frac{295869955}{161350336}a^{4}+\frac{4662475}{1646432}a^{2}-\frac{81501937}{10084396}$, $\frac{70339897681}{6763806085120}a^{17}-\frac{4763384931}{422737880320}a^{16}-\frac{24875115529}{483129006080}a^{15}+\frac{802388287}{15097781440}a^{14}+\frac{75169588709}{147039262720}a^{13}-\frac{5029511569}{9189953920}a^{12}-\frac{2380066077227}{241564503040}a^{11}+\frac{39724513123}{3774445360}a^{10}+\frac{164884471984041}{6763806085120}a^{9}-\frac{10257406113771}{422737880320}a^{8}+\frac{1599119536657}{21005608960}a^{7}-\frac{56377725461}{656425280}a^{6}+\frac{39816290928329}{422737880320}a^{5}-\frac{11032796299381}{105684470080}a^{4}-\frac{102897047865}{6039112576}a^{3}-\frac{19951218641}{754889072}a^{2}-\frac{1790424356799}{5284223504}a+\frac{285145388663}{660527938}$, $\frac{450962913}{3381903042560}a^{17}-\frac{48923333}{69018429440}a^{16}-\frac{11000609}{48312900608}a^{15}+\frac{932697933}{241564503040}a^{14}+\frac{146518189}{73519631360}a^{13}-\frac{348546719}{10502804480}a^{12}-\frac{10972553007}{120782251520}a^{11}+\frac{80316696919}{120782251520}a^{10}-\frac{147227216251}{676380608512}a^{9}-\frac{870872060011}{483129006080}a^{8}+\frac{9101417033}{2100560896}a^{7}-\frac{83162759989}{10502804480}a^{6}-\frac{328785080003}{105684470080}a^{5}+\frac{323321050901}{30195562880}a^{4}-\frac{217958319667}{15097781440}a^{3}+\frac{50342854509}{3019556288}a^{2}+\frac{40952723353}{2642111752}a-\frac{7401328483}{377444536}$, $\frac{52285504359}{6763806085120}a^{17}+\frac{3751338519}{422737880320}a^{16}-\frac{2511080953}{69018429440}a^{15}-\frac{636432533}{15097781440}a^{14}+\frac{54568611571}{147039262720}a^{13}+\frac{3936972341}{9189953920}a^{12}-\frac{1747062717853}{241564503040}a^{11}-\frac{3909347279}{471805670}a^{10}+\frac{111085381178959}{6763806085120}a^{9}+\frac{7943410397119}{422737880320}a^{8}+\frac{1296185005063}{21005608960}a^{7}+\frac{46506547259}{656425280}a^{6}+\frac{32676788048071}{422737880320}a^{5}+\frac{9211198701189}{105684470080}a^{4}-\frac{35055873983}{6039112576}a^{3}-\frac{867831261}{107841296}a^{2}-\frac{1607176859737}{5284223504}a-\frac{227666571243}{660527938}$, $\frac{134857199}{1690951521280}a^{17}+\frac{2534984523}{3381903042560}a^{16}+\frac{16180849}{17254607360}a^{15}-\frac{100421}{34509214720}a^{14}+\frac{28338627}{7351963136}a^{13}+\frac{3426889287}{73519631360}a^{12}-\frac{2170583}{12078225152}a^{11}-\frac{58598424601}{120782251520}a^{10}-\frac{1139199812377}{1690951521280}a^{9}-\frac{260528650477}{3381903042560}a^{8}+\frac{432895681}{5251402240}a^{7}-\frac{14634009669}{10502804480}a^{6}+\frac{151965258237}{42273788032}a^{5}+\frac{49359645207}{211368940160}a^{4}-\frac{398306821}{1887222680}a^{3}+\frac{28149904229}{3019556288}a^{2}-\frac{5445166505}{1321055876}a-\frac{20172285261}{2642111752}$ 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:  \( 366199805.276 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 366199805.276 \cdot 3}{2\cdot\sqrt{9254651051409408000000000000}}\cr\approx \mathstrut & 87.1461407442 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200)
 
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 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200, 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 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200);
 
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 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200);
 
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

$C_3:S_3$ (as 18T4):

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 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-2}) \), 3.1.16200.1 x3, 3.1.648.1 x3, 3.1.200.1 x3, 3.1.16200.2 x3, 6.0.2099520000.2, 6.0.3359232.4, 6.0.320000.1, 6.0.2099520000.1, 9.1.34012224000000.6 x9

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: 9.1.34012224000000.6
Minimal sibling: 9.1.34012224000000.6

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 ${\href{/padicField/7.2.0.1}{2} }^{9}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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

# 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 2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
\(3\) Copy content Toggle raw display 3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
\(5\) 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}$