↑

Properties

Label 18.10.1679341904...8125.1
Degree $18$
Signature $[10, 4]$
Discriminant $5^{9}\cdot 7^{12}\cdot 41\cdot 389246193841^{2}$
Root discriminant $195.11$
Ramified primes $5, 7, 41, 389246193841$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T857

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23276366329, 76119957655, 105636350706, 77749789756, 28793196055, 1266037761, -3366477940, -1189453818, -11020952, 79559527, 13386840, -1879383, -672908, 3231, 15408, 521, -184, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 184*x^16 + 521*x^15 + 15408*x^14 + 3231*x^13 - 672908*x^12 - 1879383*x^11 + 13386840*x^10 + 79559527*x^9 - 11020952*x^8 - 1189453818*x^7 - 3366477940*x^6 + 1266037761*x^5 + 28793196055*x^4 + 77749789756*x^3 + 105636350706*x^2 + 76119957655*x + 23276366329)
 
gp: K = bnfinit(x^18 - 6*x^17 - 184*x^16 + 521*x^15 + 15408*x^14 + 3231*x^13 - 672908*x^12 - 1879383*x^11 + 13386840*x^10 + 79559527*x^9 - 11020952*x^8 - 1189453818*x^7 - 3366477940*x^6 + 1266037761*x^5 + 28793196055*x^4 + 77749789756*x^3 + 105636350706*x^2 + 76119957655*x + 23276366329, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 184 x^{16} + 521 x^{15} + 15408 x^{14} + 3231 x^{13} - 672908 x^{12} - 1879383 x^{11} + 13386840 x^{10} + 79559527 x^{9} - 11020952 x^{8} - 1189453818 x^{7} - 3366477940 x^{6} + 1266037761 x^{5} + 28793196055 x^{4} + 77749789756 x^{3} + 105636350706 x^{2} + 76119957655 x + 23276366329 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(167934190485676783463200582052921330078125=5^{9}\cdot 7^{12}\cdot 41\cdot 389246193841^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.11$
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:  $5, 7, 41, 389246193841$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{1}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7}$, $\frac{1}{25000104533454670368168908768974046425347029021154881917427} a^{17} - \frac{1627553366788049640881868673672029474794419566674939388178}{25000104533454670368168908768974046425347029021154881917427} a^{16} + \frac{8464491642436846403943549577487119467693991538585675852525}{25000104533454670368168908768974046425347029021154881917427} a^{15} - \frac{710878409472493242075967679244319249161460975995421566920}{25000104533454670368168908768974046425347029021154881917427} a^{14} - \frac{8423611136334091132599339749453845964577241783494359812707}{25000104533454670368168908768974046425347029021154881917427} a^{13} - \frac{7393816420089435490892986763247394557328695485071492287281}{25000104533454670368168908768974046425347029021154881917427} a^{12} + \frac{5015469678086256683931346112030530713292614746377610828131}{25000104533454670368168908768974046425347029021154881917427} a^{11} + \frac{8812766247232489796130239573268950014902275259016625892305}{25000104533454670368168908768974046425347029021154881917427} a^{10} + \frac{500161559226314768120227900572495139201155808982743003043}{25000104533454670368168908768974046425347029021154881917427} a^{9} + \frac{858091322773579198724132449624824240259304335521449322928}{25000104533454670368168908768974046425347029021154881917427} a^{8} + \frac{6446426817719155611425647539613981739495948305235672451573}{25000104533454670368168908768974046425347029021154881917427} a^{7} + \frac{1244987801072067586141063958243965507485997856719375612453}{25000104533454670368168908768974046425347029021154881917427} a^{6} - \frac{7805020364814528619777013956152287196569724938420653384394}{25000104533454670368168908768974046425347029021154881917427} a^{5} - \frac{8232640335218778756547999165517055504974291993211819254652}{25000104533454670368168908768974046425347029021154881917427} a^{4} - \frac{11614789002080999396143987282108918491836775315038192449875}{25000104533454670368168908768974046425347029021154881917427} a^{3} + \frac{440200632042864984737056408518347306801030884770848940447}{3571443504779238624024129824139149489335289860164983131061} a^{2} - \frac{1096948204801319447313204951429889116076689922504435981189}{25000104533454670368168908768974046425347029021154881917427} a + \frac{8379416217282421772532275684495824227151082384842805885007}{25000104533454670368168908768974046425347029021154881917427}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $13$
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:  \( 38237156620700 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 6.6.300125.1

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

Sibling fields

Degree 18 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
5Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
41Data not computed
389246193841Data not computed