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Properties

Label 18.0.22030616752...4016.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 7^{12}\cdot 17^{9}$
Root discriminant $42.67$
Ramified primes $2, 7, 17$
Class number $112$ (GRH)
Class group $[2, 2, 28]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42875, 17150, 60025, 19698, 75068, 13328, 50645, 5572, 25921, 62, 9401, -490, 2049, -98, 294, -6, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 28*x^16 - 6*x^15 + 294*x^14 - 98*x^13 + 2049*x^12 - 490*x^11 + 9401*x^10 + 62*x^9 + 25921*x^8 + 5572*x^7 + 50645*x^6 + 13328*x^5 + 75068*x^4 + 19698*x^3 + 60025*x^2 + 17150*x + 42875)
 
gp: K = bnfinit(x^18 + 28*x^16 - 6*x^15 + 294*x^14 - 98*x^13 + 2049*x^12 - 490*x^11 + 9401*x^10 + 62*x^9 + 25921*x^8 + 5572*x^7 + 50645*x^6 + 13328*x^5 + 75068*x^4 + 19698*x^3 + 60025*x^2 + 17150*x + 42875, 1)
 

Normalized defining polynomial

\( x^{18} + 28 x^{16} - 6 x^{15} + 294 x^{14} - 98 x^{13} + 2049 x^{12} - 490 x^{11} + 9401 x^{10} + 62 x^{9} + 25921 x^{8} + 5572 x^{7} + 50645 x^{6} + 13328 x^{5} + 75068 x^{4} + 19698 x^{3} + 60025 x^{2} + 17150 x + 42875 \)

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:  \(-220306167525977029404855894016=-\,2^{27}\cdot 7^{12}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.67$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{28} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} + \frac{2}{7} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{5}{28} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{2}{7} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{812} a^{13} - \frac{1}{203} a^{12} + \frac{11}{116} a^{11} - \frac{17}{406} a^{10} + \frac{241}{812} a^{9} + \frac{12}{29} a^{8} + \frac{369}{812} a^{7} - \frac{89}{203} a^{6} + \frac{55}{116} a^{5} - \frac{123}{406} a^{4} - \frac{3}{28} a^{3} + \frac{13}{29} a^{2} - \frac{1}{4} a - \frac{25}{58}$, $\frac{1}{812} a^{14} + \frac{3}{812} a^{12} + \frac{137}{406} a^{11} - \frac{43}{116} a^{10} + \frac{6}{203} a^{9} - \frac{317}{812} a^{8} + \frac{11}{29} a^{7} + \frac{295}{812} a^{6} - \frac{165}{406} a^{5} + \frac{21}{116} a^{4} - \frac{83}{203} a^{3} + \frac{5}{116} a^{2} - \frac{25}{58} a - \frac{13}{58}$, $\frac{1}{619556} a^{15} + \frac{10}{22127} a^{13} + \frac{2269}{619556} a^{12} - \frac{585}{6322} a^{11} + \frac{31515}{88508} a^{10} - \frac{34836}{154889} a^{9} - \frac{121}{436} a^{8} - \frac{4597}{44254} a^{7} + \frac{47305}{619556} a^{6} + \frac{13}{58} a^{5} - \frac{25413}{88508} a^{4} - \frac{180}{22127} a^{3} - \frac{1939}{12644} a^{2} - \frac{2883}{12644} a + \frac{4041}{12644}$, $\frac{1}{22548740620} a^{16} - \frac{723}{1127437031} a^{15} - \frac{1953131}{3221248660} a^{14} - \frac{12180111}{22548740620} a^{13} - \frac{36260131}{22548740620} a^{12} - \frac{1065549879}{3221248660} a^{11} - \frac{3737005281}{22548740620} a^{10} - \frac{1487785073}{4509748124} a^{9} + \frac{144251663}{3221248660} a^{8} + \frac{7870853997}{22548740620} a^{7} - \frac{8948963039}{22548740620} a^{6} - \frac{67838597}{460178380} a^{5} + \frac{41136621}{644249732} a^{4} + \frac{1467054879}{3221248660} a^{3} + \frac{1450173}{115044595} a^{2} - \frac{5306077}{15868220} a - \frac{6477986}{23008919}$, $\frac{1}{7057158159689866900} a^{17} + \frac{5568617}{1411431631937973380} a^{16} + \frac{1298175154207}{1764289539922466725} a^{15} - \frac{2592768243578701}{7057158159689866900} a^{14} - \frac{4072384096428791}{7057158159689866900} a^{13} - \frac{39318967739400179}{3528579079844933450} a^{12} + \frac{1192719259672111869}{7057158159689866900} a^{11} - \frac{64431420734490041}{141143163193797338} a^{10} + \frac{3071756837557412451}{7057158159689866900} a^{9} - \frac{912053174352164289}{3528579079844933450} a^{8} - \frac{1771080558841771909}{7057158159689866900} a^{7} + \frac{460149732608884141}{3528579079844933450} a^{6} - \frac{19855949877204461}{201633090276853340} a^{5} + \frac{6477511867635243}{36005908978009525} a^{4} + \frac{41215053476765241}{252041362846066675} a^{3} - \frac{2449666564210007}{4966332272828900} a^{2} - \frac{2323185432352631}{28804727182407620} a + \frac{60399171858409}{198653290913156}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{28}$, which has order $112$ (assuming GRH)

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:  \( -1 \) (order $2$)
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 194089.946384 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-34}) \), 3.1.6664.1 x3, \(\Q(\zeta_{7})^+\), 6.0.6039609856.1, 6.0.123257344.1 x2, 6.0.6039609856.5, 9.3.295940882944.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.123257344.1
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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

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
$2$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$