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Properties

Label 18.0.29093534383...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 7^{9}\cdot 37^{14}$
Root discriminant $155.75$
Ramified primes $2, 5, 7, 37$
Class number $2125872$ (GRH)
Class group $[2, 2, 6, 88578]$ (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![3659949451, -2720012338, 2837225215, -1401562361, 842632363, -313467091, 138893691, -41171933, 14782497, -3586552, 1098634, -219953, 59557, -9925, 2471, -346, 75, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 75*x^16 - 346*x^15 + 2471*x^14 - 9925*x^13 + 59557*x^12 - 219953*x^11 + 1098634*x^10 - 3586552*x^9 + 14782497*x^8 - 41171933*x^7 + 138893691*x^6 - 313467091*x^5 + 842632363*x^4 - 1401562361*x^3 + 2837225215*x^2 - 2720012338*x + 3659949451)
 
gp: K = bnfinit(x^18 - 7*x^17 + 75*x^16 - 346*x^15 + 2471*x^14 - 9925*x^13 + 59557*x^12 - 219953*x^11 + 1098634*x^10 - 3586552*x^9 + 14782497*x^8 - 41171933*x^7 + 138893691*x^6 - 313467091*x^5 + 842632363*x^4 - 1401562361*x^3 + 2837225215*x^2 - 2720012338*x + 3659949451, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 75 x^{16} - 346 x^{15} + 2471 x^{14} - 9925 x^{13} + 59557 x^{12} - 219953 x^{11} + 1098634 x^{10} - 3586552 x^{9} + 14782497 x^{8} - 41171933 x^{7} + 138893691 x^{6} - 313467091 x^{5} + 842632363 x^{4} - 1401562361 x^{3} + 2837225215 x^{2} - 2720012338 x + 3659949451 \)

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:  \(-2909353438387945911099105025784000000000=-\,2^{12}\cdot 5^{9}\cdot 7^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.75$
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:  $2, 5, 7, 37$
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}{49034065505992864346348302007219547021935181911371567869} a^{17} + \frac{1356004673448156318115224739000886101234873038741913399}{16344688501997621448782767335739849007311727303790522623} a^{16} - \frac{2225212961378692711704635975077652905362485162758652792}{16344688501997621448782767335739849007311727303790522623} a^{15} + \frac{2158323294224443552859090753887258845614115046689790266}{16344688501997621448782767335739849007311727303790522623} a^{14} - \frac{6635329099345878027518314220671332266440887247502427490}{49034065505992864346348302007219547021935181911371567869} a^{13} + \frac{1032222080776381171472819180447191842595876634363795757}{16344688501997621448782767335739849007311727303790522623} a^{12} - \frac{842906559394430888664950684149464017745764225141655401}{16344688501997621448782767335739849007311727303790522623} a^{11} - \frac{785083333470546226376053595252176407997413022977560642}{1581744048580414978914461355071598291030167158431340899} a^{10} + \frac{11246134251070267957904080939396190833188422053812161606}{49034065505992864346348302007219547021935181911371567869} a^{9} - \frac{18888428422283078699350636956132470779264471853446036570}{49034065505992864346348302007219547021935181911371567869} a^{8} - \frac{22160088448602637645752977557048360903918684838175791683}{49034065505992864346348302007219547021935181911371567869} a^{7} + \frac{112969645734838926748601892714115196615660616683895126}{527248016193471659638153785023866097010055719477113633} a^{6} + \frac{12257465335401298413139932627240999212163934974771634222}{49034065505992864346348302007219547021935181911371567869} a^{5} - \frac{2170964131686699145374900818530559717924296629923564908}{49034065505992864346348302007219547021935181911371567869} a^{4} + \frac{3391282889200816588366029653248458582125338611619037486}{16344688501997621448782767335739849007311727303790522623} a^{3} + \frac{22308450428436073843623634814356347079768555897874239013}{49034065505992864346348302007219547021935181911371567869} a^{2} + \frac{7050313788533775670502040327710680431414029482942401809}{16344688501997621448782767335739849007311727303790522623} a - \frac{843721852946100296060092042079882401011137634896316869}{2131915891564907145493404435096502044431964430929198603}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{88578}$, which has order $2125872$ (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

\(\Q(\sqrt{-35}) \), 3.3.1369.1, 3.3.148.1, 6.0.80354652875.4, 6.0.939134000.6, 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 R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$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}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$