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Properties

Label 18.0.18976385690...6247.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{17}\cdot 13^{8}$
Root discriminant $19.64$
Ramified primes $7, 13$
Class number $1$
Class group Trivial
Galois group $C_2\times C_9:C_3$ (as 18T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![421, -2458, 5988, -7270, 3474, 1533, -2352, 213, 529, 373, -586, 6, 259, -91, -24, 31, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 2*x^16 + 31*x^15 - 24*x^14 - 91*x^13 + 259*x^12 + 6*x^11 - 586*x^10 + 373*x^9 + 529*x^8 + 213*x^7 - 2352*x^6 + 1533*x^5 + 3474*x^4 - 7270*x^3 + 5988*x^2 - 2458*x + 421)
 
gp: K = bnfinit(x^18 - 2*x^17 - 2*x^16 + 31*x^15 - 24*x^14 - 91*x^13 + 259*x^12 + 6*x^11 - 586*x^10 + 373*x^9 + 529*x^8 + 213*x^7 - 2352*x^6 + 1533*x^5 + 3474*x^4 - 7270*x^3 + 5988*x^2 - 2458*x + 421, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 2 x^{16} + 31 x^{15} - 24 x^{14} - 91 x^{13} + 259 x^{12} + 6 x^{11} - 586 x^{10} + 373 x^{9} + 529 x^{8} + 213 x^{7} - 2352 x^{6} + 1533 x^{5} + 3474 x^{4} - 7270 x^{3} + 5988 x^{2} - 2458 x + 421 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-189763856901384950886247=-\,7^{17}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.64$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{281} a^{16} - \frac{104}{281} a^{15} - \frac{12}{281} a^{14} + \frac{73}{281} a^{13} - \frac{41}{281} a^{12} + \frac{41}{281} a^{11} + \frac{80}{281} a^{10} - \frac{74}{281} a^{9} - \frac{40}{281} a^{8} + \frac{13}{281} a^{7} - \frac{106}{281} a^{6} + \frac{3}{281} a^{5} - \frac{26}{281} a^{4} - \frac{131}{281} a^{3} + \frac{102}{281} a^{2} + \frac{37}{281} a - \frac{96}{281}$, $\frac{1}{1173254722973313716373877} a^{17} - \frac{675413422919748185190}{1173254722973313716373877} a^{16} - \frac{403479096640902310219665}{1173254722973313716373877} a^{15} - \frac{277497479974840650635642}{1173254722973313716373877} a^{14} - \frac{245992867974139512361784}{1173254722973313716373877} a^{13} + \frac{61726025453922707121171}{1173254722973313716373877} a^{12} - \frac{262707826410628155333850}{1173254722973313716373877} a^{11} - \frac{176973769115643526435517}{1173254722973313716373877} a^{10} - \frac{381669271086923433949166}{1173254722973313716373877} a^{9} - \frac{195189430822730080707237}{1173254722973313716373877} a^{8} + \frac{321510283324791758689733}{1173254722973313716373877} a^{7} - \frac{253297213964833257904371}{1173254722973313716373877} a^{6} - \frac{114048136713588229306587}{1173254722973313716373877} a^{5} + \frac{335890794784597605354875}{1173254722973313716373877} a^{4} + \frac{341134888049748248110029}{1173254722973313716373877} a^{3} + \frac{424723996960910138226321}{1173254722973313716373877} a^{2} + \frac{7498053323431526937681}{1173254722973313716373877} a + \frac{308993523260833349445615}{1173254722973313716373877}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2544108032611958202284}{40457059412872886771513} a^{17} + \frac{3871203719646744047979}{40457059412872886771513} a^{16} + \frac{7072424679893812006966}{40457059412872886771513} a^{15} - \frac{75680469856564288359190}{40457059412872886771513} a^{14} + \frac{24464196631149971900818}{40457059412872886771513} a^{13} + \frac{247001869987483801768213}{40457059412872886771513} a^{12} - \frac{541658120904277198840609}{40457059412872886771513} a^{11} - \frac{287578220970904329044373}{40457059412872886771513} a^{10} + \frac{1377240764222741705324960}{40457059412872886771513} a^{9} - \frac{272657816027231189743704}{40457059412872886771513} a^{8} - \frac{1543976052306633268970578}{40457059412872886771513} a^{7} - \frac{1291631756108964218465412}{40457059412872886771513} a^{6} + \frac{5442934761572288632844239}{40457059412872886771513} a^{5} - \frac{1209077575384908804865726}{40457059412872886771513} a^{4} - \frac{9683468428079253771891191}{40457059412872886771513} a^{3} + \frac{13876744204672193332444380}{40457059412872886771513} a^{2} - \frac{8135302318489322144843390}{40457059412872886771513} a + \frac{1866344539016255139380778}{40457059412872886771513} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 89545.2892541 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_9:C_3$ (as 18T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times C_9:C_3$
Character table for $C_2\times C_9:C_3$ is not computed

Intermediate fields

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

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
$13$13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$