Properties

Label 18.0.14929107797...1783.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,23^{9}\cdot 2879^{2}$
Root discriminant $11.62$
Ramified primes $23, 2879$
Class number $1$
Class group Trivial
Galois Group 18T319

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 2, 1, 5, -5, 8, -4, 6, 5, 5, 4, -5, 2, 3, 1, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - x^16 + x^15 + 3*x^14 + 2*x^13 - 5*x^12 + 4*x^11 + 5*x^10 + 5*x^9 + 6*x^8 - 4*x^7 + 8*x^6 - 5*x^5 + 5*x^4 + x^3 + 2*x^2 - x + 1)
gp: K = bnfinit(x^18 - x^17 - x^16 + x^15 + 3*x^14 + 2*x^13 - 5*x^12 + 4*x^11 + 5*x^10 + 5*x^9 + 6*x^8 - 4*x^7 + 8*x^6 - 5*x^5 + 5*x^4 + x^3 + 2*x^2 - x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut x^{17} \) \(\mathstrut -\mathstrut x^{16} \) \(\mathstrut +\mathstrut x^{15} \) \(\mathstrut +\mathstrut 3 x^{14} \) \(\mathstrut +\mathstrut 2 x^{13} \) \(\mathstrut -\mathstrut 5 x^{12} \) \(\mathstrut +\mathstrut 4 x^{11} \) \(\mathstrut +\mathstrut 5 x^{10} \) \(\mathstrut +\mathstrut 5 x^{9} \) \(\mathstrut +\mathstrut 6 x^{8} \) \(\mathstrut -\mathstrut 4 x^{7} \) \(\mathstrut +\mathstrut 8 x^{6} \) \(\mathstrut -\mathstrut 5 x^{5} \) \(\mathstrut +\mathstrut 5 x^{4} \) \(\mathstrut +\mathstrut x^{3} \) \(\mathstrut +\mathstrut 2 x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-14929107797061341783=-\,23^{9}\cdot 2879^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.62$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $23, 2879$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{33} a^{16} + \frac{4}{33} a^{15} - \frac{2}{33} a^{14} - \frac{5}{33} a^{13} - \frac{2}{33} a^{12} - \frac{13}{33} a^{11} + \frac{5}{33} a^{10} + \frac{16}{33} a^{9} + \frac{2}{33} a^{8} + \frac{3}{11} a^{7} - \frac{13}{33} a^{6} + \frac{2}{11} a^{5} + \frac{1}{11} a^{4} + \frac{5}{33} a^{3} + \frac{1}{3} a^{2} - \frac{16}{33} a + \frac{10}{33}$, $\frac{1}{1693593} a^{17} - \frac{47}{17107} a^{16} + \frac{142256}{1693593} a^{15} + \frac{23852}{188177} a^{14} + \frac{40816}{564531} a^{13} - \frac{324241}{1693593} a^{12} + \frac{4993}{188177} a^{11} - \frac{67808}{1693593} a^{10} + \frac{247483}{564531} a^{9} + \frac{800372}{1693593} a^{8} + \frac{777827}{1693593} a^{7} + \frac{295144}{1693593} a^{6} + \frac{10718}{564531} a^{5} + \frac{276907}{1693593} a^{4} - \frac{210037}{564531} a^{3} + \frac{539347}{1693593} a^{2} - \frac{277003}{564531} a + \frac{202184}{1693593}$

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

Class group and class number

Trivial Abelian group, 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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{631658}{1693593} a^{17} - \frac{223166}{564531} a^{16} - \frac{67921}{153963} a^{15} + \frac{275650}{564531} a^{14} + \frac{221895}{188177} a^{13} + \frac{1193518}{1693593} a^{12} - \frac{1285831}{564531} a^{11} + \frac{2016035}{1693593} a^{10} + \frac{1364024}{564531} a^{9} + \frac{3081271}{1693593} a^{8} + \frac{3005383}{1693593} a^{7} - \frac{4703047}{1693593} a^{6} + \frac{1486396}{564531} a^{5} - \frac{1351147}{1693593} a^{4} + \frac{219364}{188177} a^{3} + \frac{1373039}{1693593} a^{2} - \frac{320192}{564531} a - \frac{500255}{1693593} \),  \( \frac{447739}{1693593} a^{17} - \frac{87166}{564531} a^{16} - \frac{38246}{153963} a^{15} - \frac{1797}{188177} a^{14} + \frac{379381}{564531} a^{13} + \frac{1811627}{1693593} a^{12} - \frac{287939}{564531} a^{11} + \frac{1063525}{1693593} a^{10} + \frac{81161}{188177} a^{9} + \frac{3539024}{1693593} a^{8} + \frac{5540957}{1693593} a^{7} - \frac{92546}{1693593} a^{6} + \frac{208938}{188177} a^{5} - \frac{4340627}{1693593} a^{4} + \frac{742934}{564531} a^{3} + \frac{83218}{1693593} a^{2} + \frac{618248}{564531} a + \frac{909323}{1693593} \),  \( \frac{344261}{1693593} a^{17} - \frac{8242}{51321} a^{16} - \frac{435965}{1693593} a^{15} + \frac{21800}{188177} a^{14} + \frac{368563}{564531} a^{13} + \frac{1183729}{1693593} a^{12} - \frac{493343}{564531} a^{11} + \frac{271493}{1693593} a^{10} + \frac{479251}{564531} a^{9} + \frac{2832736}{1693593} a^{8} + \frac{3205210}{1693593} a^{7} - \frac{2237044}{1693593} a^{6} + \frac{391136}{564531} a^{5} - \frac{682057}{1693593} a^{4} + \frac{197885}{188177} a^{3} + \frac{1327136}{1693593} a^{2} + \frac{375799}{564531} a + \frac{780910}{1693593} \),  \( \frac{18500}{188177} a^{17} - \frac{49397}{188177} a^{16} - \frac{100175}{564531} a^{15} + \frac{194669}{564531} a^{14} + \frac{279320}{564531} a^{13} - \frac{214274}{564531} a^{12} - \frac{288091}{188177} a^{11} + \frac{144679}{564531} a^{10} + \frac{61986}{188177} a^{9} - \frac{133171}{188177} a^{8} - \frac{868708}{564531} a^{7} - \frac{1650196}{564531} a^{6} - \frac{75005}{51321} a^{5} - \frac{424860}{188177} a^{4} + \frac{166390}{564531} a^{3} + \frac{22252}{188177} a^{2} - \frac{82718}{564531} a - \frac{212620}{188177} \),  \( \frac{500255}{1693593} a^{17} + \frac{43801}{564531} a^{16} - \frac{1169753}{1693593} a^{15} - \frac{82292}{564531} a^{14} + \frac{258635}{188177} a^{13} + \frac{2997565}{1693593} a^{12} - \frac{39629}{51321} a^{11} - \frac{1856473}{1693593} a^{10} + \frac{1505770}{564531} a^{9} + \frac{6593347}{1693593} a^{8} + \frac{6082801}{1693593} a^{7} + \frac{1004363}{1693593} a^{6} - \frac{233669}{564531} a^{5} + \frac{1957913}{1693593} a^{4} + \frac{127792}{188177} a^{3} + \frac{2474531}{1693593} a^{2} + \frac{791183}{564531} a + \frac{232762}{1693593} \),  \( \frac{12775}{153963} a^{17} - \frac{113737}{564531} a^{16} - \frac{339446}{1693593} a^{15} + \frac{117584}{564531} a^{14} + \frac{375580}{564531} a^{13} - \frac{258947}{1693593} a^{12} - \frac{1016153}{564531} a^{11} - \frac{462340}{1693593} a^{10} + \frac{591419}{564531} a^{9} + \frac{918133}{1693593} a^{8} - \frac{3894194}{1693593} a^{7} - \frac{6781243}{1693593} a^{6} - \frac{432326}{564531} a^{5} - \frac{816976}{1693593} a^{4} + \frac{711770}{564531} a^{3} - \frac{148214}{153963} a^{2} - \frac{151100}{564531} a - \frac{168815}{1693593} \),  \( \frac{493513}{1693593} a^{17} - \frac{80851}{188177} a^{16} - \frac{17978}{153963} a^{15} + \frac{65125}{188177} a^{14} + \frac{418388}{564531} a^{13} + \frac{446342}{1693593} a^{12} - \frac{903605}{564531} a^{11} + \frac{2859127}{1693593} a^{10} + \frac{326246}{564531} a^{9} + \frac{2218262}{1693593} a^{8} + \frac{2052551}{1693593} a^{7} - \frac{3862847}{1693593} a^{6} + \frac{577616}{188177} a^{5} - \frac{4408868}{1693593} a^{4} + \frac{488144}{188177} a^{3} - \frac{1045958}{1693593} a^{2} + \frac{258073}{188177} a - \frac{627142}{1693593} \),  \( \frac{962}{564531} a^{17} + \frac{211132}{564531} a^{16} - \frac{211012}{564531} a^{15} - \frac{87231}{188177} a^{14} + \frac{89962}{188177} a^{13} + \frac{676369}{564531} a^{12} + \frac{359252}{564531} a^{11} - \frac{1148474}{564531} a^{10} + \frac{257132}{188177} a^{9} + \frac{1220605}{564531} a^{8} + \frac{288248}{188177} a^{7} + \frac{440296}{188177} a^{6} - \frac{1160584}{564531} a^{5} + \frac{1191820}{564531} a^{4} - \frac{26876}{17107} a^{3} + \frac{1176887}{564531} a^{2} + \frac{218950}{564531} a + \frac{29041}{51321} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 92.6925275389 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T319:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1296
The 22 conjugacy class representatives for t18n319
Character table for t18n319 is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.35028793.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
2879Data not computed