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Properties

Label 18.0.23264650494...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 19^{9}\cdot 37^{14}$
Root discriminant $256.60$
Ramified primes $2, 5, 19, 37$
Class number $236411136$ (GRH)
Class group $[2, 2, 6, 12, 820872]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6965060119081, -2062907686453, 2204925392350, -513902613356, 303026115883, -57241530331, 24313306146, -3772636223, 1278014562, -163118347, 46317994, -4806923, 1171102, -95635, 20051, -1186, 210, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 210*x^16 - 1186*x^15 + 20051*x^14 - 95635*x^13 + 1171102*x^12 - 4806923*x^11 + 46317994*x^10 - 163118347*x^9 + 1278014562*x^8 - 3772636223*x^7 + 24313306146*x^6 - 57241530331*x^5 + 303026115883*x^4 - 513902613356*x^3 + 2204925392350*x^2 - 2062907686453*x + 6965060119081)
 
gp: K = bnfinit(x^18 - 7*x^17 + 210*x^16 - 1186*x^15 + 20051*x^14 - 95635*x^13 + 1171102*x^12 - 4806923*x^11 + 46317994*x^10 - 163118347*x^9 + 1278014562*x^8 - 3772636223*x^7 + 24313306146*x^6 - 57241530331*x^5 + 303026115883*x^4 - 513902613356*x^3 + 2204925392350*x^2 - 2062907686453*x + 6965060119081, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 210 x^{16} - 1186 x^{15} + 20051 x^{14} - 95635 x^{13} + 1171102 x^{12} - 4806923 x^{11} + 46317994 x^{10} - 163118347 x^{9} + 1278014562 x^{8} - 3772636223 x^{7} + 24313306146 x^{6} - 57241530331 x^{5} + 303026115883 x^{4} - 513902613356 x^{3} + 2204925392350 x^{2} - 2062907686453 x + 6965060119081 \)

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:  \(-23264650494782882410421125533525848000000000=-\,2^{12}\cdot 5^{9}\cdot 19^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $256.60$
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:  $2, 5, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{17} + \frac{204295637438092635762028174754535836080316298073346823206532845355191}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{16} + \frac{882389008722369795622469247081593169725974418013905615234668551929147}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{15} - \frac{39100371126197010087229097670965884831575920363037412437531045567809}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{14} + \frac{860079754026636032982053621789589882332296888058937763817539495114813}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{13} + \frac{923534264391341077419438423775173975792287439691718441122546392328334}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{12} - \frac{34499656099555775781985109771422110808913796077966299768195375875338}{2278463420681137276676861262149653549249755181700776187351104271378843} a^{11} - \frac{1552513654891391803042924313087615471018226396052593433172861267387374}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{10} - \frac{183801523067480552606022618930167718395312426484630089963136336916061}{2278463420681137276676861262149653549249755181700776187351104271378843} a^{9} + \frac{1632841180438677537837732929603726381815085278636709274051053026748615}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{8} - \frac{704849068575832244498561951061476574864630030583276789523832634767677}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{7} - \frac{2163294812067800380563431793900488826461844992164964072837558889590378}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{6} + \frac{3160212872409911279543512340235523711124459641790749062884973889667189}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{5} - \frac{1630264824549743110235561088364500740008972492921376959673984779493573}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{4} + \frac{904451826459100518679682644050920159639875597481554886006513420291964}{2278463420681137276676861262149653549249755181700776187351104271378843} a^{3} - \frac{852191508454344621245397448390132206312872653631204853877311241505246}{6835390262043411830030583786448960647749265545102328562053312814136529} a^{2} + \frac{689038322542321528397420058636308418273265920472575922568248372465308}{2278463420681137276676861262149653549249755181700776187351104271378843} a + \frac{612695738631872483377629435124637756632728253949775094293386584599952}{2278463420681137276676861262149653549249755181700776187351104271378843}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{12}\times C_{820872}$, which has order $236411136$ (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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-95}) \), 3.3.148.1, 3.3.1369.1, 6.0.18779942000.2, 6.0.1606858787375.3, 9.9.6075640136512.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$