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Properties

Label 18.0.41963408025...9703.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,13^{12}\cdot 23^{9}$
Root discriminant $26.52$
Ramified primes $13, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15625, -5250, 37718, 13096, 28737, 27725, 15289, 15264, 7549, 2973, 2550, -244, 776, -252, 205, -62, 24, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 24*x^16 - 62*x^15 + 205*x^14 - 252*x^13 + 776*x^12 - 244*x^11 + 2550*x^10 + 2973*x^9 + 7549*x^8 + 15264*x^7 + 15289*x^6 + 27725*x^5 + 28737*x^4 + 13096*x^3 + 37718*x^2 - 5250*x + 15625)
 
gp: K = bnfinit(x^18 - 5*x^17 + 24*x^16 - 62*x^15 + 205*x^14 - 252*x^13 + 776*x^12 - 244*x^11 + 2550*x^10 + 2973*x^9 + 7549*x^8 + 15264*x^7 + 15289*x^6 + 27725*x^5 + 28737*x^4 + 13096*x^3 + 37718*x^2 - 5250*x + 15625, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 24 x^{16} - 62 x^{15} + 205 x^{14} - 252 x^{13} + 776 x^{12} - 244 x^{11} + 2550 x^{10} + 2973 x^{9} + 7549 x^{8} + 15264 x^{7} + 15289 x^{6} + 27725 x^{5} + 28737 x^{4} + 13096 x^{3} + 37718 x^{2} - 5250 x + 15625 \)

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:  \(-41963408025348177483649703=-\,13^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.52$
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:  $13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{115} a^{13} - \frac{47}{115} a^{12} - \frac{8}{23} a^{11} + \frac{22}{115} a^{10} - \frac{56}{115} a^{9} + \frac{36}{115} a^{8} - \frac{27}{115} a^{7} + \frac{2}{115} a^{6} - \frac{29}{115} a^{5} - \frac{2}{115} a^{4} + \frac{53}{115} a^{3} + \frac{7}{23} a^{2} - \frac{47}{115} a + \frac{1}{23}$, $\frac{1}{115} a^{14} + \frac{51}{115} a^{12} - \frac{18}{115} a^{11} - \frac{57}{115} a^{10} + \frac{49}{115} a^{9} + \frac{11}{23} a^{8} - \frac{2}{115} a^{7} - \frac{10}{23} a^{6} + \frac{3}{23} a^{5} - \frac{41}{115} a^{4} - \frac{4}{115} a^{3} - \frac{12}{115} a^{2} - \frac{19}{115} a + \frac{1}{23}$, $\frac{1}{44275} a^{15} + \frac{64}{44275} a^{14} + \frac{4}{4025} a^{13} - \frac{72}{1265} a^{12} + \frac{8501}{44275} a^{11} + \frac{6712}{44275} a^{10} + \frac{19}{6325} a^{9} + \frac{597}{1925} a^{8} - \frac{6429}{44275} a^{7} - \frac{3429}{44275} a^{6} + \frac{21132}{44275} a^{5} - \frac{4569}{44275} a^{4} + \frac{153}{575} a^{3} - \frac{13912}{44275} a^{2} + \frac{1369}{6325} a - \frac{674}{1771}$, $\frac{1}{60081175} a^{16} + \frac{24}{2403247} a^{15} + \frac{97103}{60081175} a^{14} + \frac{207019}{60081175} a^{13} - \frac{15116749}{60081175} a^{12} + \frac{245858}{60081175} a^{11} + \frac{4178393}{12016235} a^{10} - \frac{2285506}{5461925} a^{9} + \frac{27416892}{60081175} a^{8} - \frac{20062893}{60081175} a^{7} - \frac{21318602}{60081175} a^{6} - \frac{937119}{2612225} a^{5} - \frac{29113468}{60081175} a^{4} + \frac{18207599}{60081175} a^{3} + \frac{3798216}{60081175} a^{2} + \frac{9134033}{60081175} a + \frac{836086}{2403247}$, $\frac{1}{18221825541400758416437375} a^{17} + \frac{29438504690286228}{3644365108280151683287475} a^{16} + \frac{18372258668668256052}{2603117934485822630919625} a^{15} - \frac{252253718797082721319}{423763384683738567824125} a^{14} + \frac{766319360041610244752}{520623586897164526183925} a^{13} + \frac{797223123904911250880188}{1656529594672796219676125} a^{12} - \frac{7685772602813517297703174}{18221825541400758416437375} a^{11} + \frac{5824876930067037853916761}{18221825541400758416437375} a^{10} + \frac{27343945470094373355551}{66261183786911848787045} a^{9} - \frac{278891846879867275594446}{2603117934485822630919625} a^{8} - \frac{8626787348576032390837646}{18221825541400758416437375} a^{7} - \frac{5071138798257542528716211}{18221825541400758416437375} a^{6} + \frac{4587375801161133113914204}{18221825541400758416437375} a^{5} + \frac{1714082832841131213227091}{3644365108280151683287475} a^{4} - \frac{180176753832791847949388}{1656529594672796219676125} a^{3} - \frac{958370089728667308812602}{2603117934485822630919625} a^{2} - \frac{91250232381233871160899}{792253284408728626801625} a + \frac{44699383184331317545266}{145774604331206067331499}$

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:  $8$
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:  \( 105070.48842 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.1350739057369.1 x9

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{18}$

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
$13$13.9.6.2$x^{9} - 338 x^{3} + 13182$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
13.9.6.2$x^{9} - 338 x^{3} + 13182$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$