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Properties

Label 18.0.21768647730...4931.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 71^{3}\cdot 181\cdot 28955681^{3}$
Root discriminant $174.18$
Ramified primes $7, 71, 181, 28955681$
Class number $3374$ (GRH)
Class group $[3374]$ (GRH)
Galois group 18T926

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![72067141, 0, 9473184, 798147, 2772355, -2306607, 826339, 103572, -142708, 44326, 8083, -9432, 2973, -4, -187, 69, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 3*x^16 + 69*x^15 - 187*x^14 - 4*x^13 + 2973*x^12 - 9432*x^11 + 8083*x^10 + 44326*x^9 - 142708*x^8 + 103572*x^7 + 826339*x^6 - 2306607*x^5 + 2772355*x^4 + 798147*x^3 + 9473184*x^2 + 72067141)
 
gp: K = bnfinit(x^18 - 2*x^17 - 3*x^16 + 69*x^15 - 187*x^14 - 4*x^13 + 2973*x^12 - 9432*x^11 + 8083*x^10 + 44326*x^9 - 142708*x^8 + 103572*x^7 + 826339*x^6 - 2306607*x^5 + 2772355*x^4 + 798147*x^3 + 9473184*x^2 + 72067141, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 3 x^{16} + 69 x^{15} - 187 x^{14} - 4 x^{13} + 2973 x^{12} - 9432 x^{11} + 8083 x^{10} + 44326 x^{9} - 142708 x^{8} + 103572 x^{7} + 826339 x^{6} - 2306607 x^{5} + 2772355 x^{4} + 798147 x^{3} + 9473184 x^{2} + 72067141 \)

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:  \(-21768647730913562905265507945105445204931=-\,7^{12}\cdot 71^{3}\cdot 181\cdot 28955681^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $174.18$
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:  $7, 71, 181, 28955681$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3062851360274591447343732929818426907400340708943550436886023} a^{17} + \frac{682504195121086751226086125155021647211252781057068652157990}{3062851360274591447343732929818426907400340708943550436886023} a^{16} - \frac{690846623180648765855210007672640607465870360299587083271156}{3062851360274591447343732929818426907400340708943550436886023} a^{15} - \frac{596395275673088583815843632895700269570498800775634715473285}{3062851360274591447343732929818426907400340708943550436886023} a^{14} - \frac{1335515453217295951322466546727421119290392371484775440333201}{3062851360274591447343732929818426907400340708943550436886023} a^{13} + \frac{1275600917015271934197911419524884892386338869416104660450481}{3062851360274591447343732929818426907400340708943550436886023} a^{12} + \frac{1146618357125147649843973804590273703758327433629847355039265}{3062851360274591447343732929818426907400340708943550436886023} a^{11} + \frac{996238529223441095751176174844241498606609979700752290005401}{3062851360274591447343732929818426907400340708943550436886023} a^{10} - \frac{1434048057151439641582376727918116793997910086873892263493817}{3062851360274591447343732929818426907400340708943550436886023} a^{9} + \frac{227090760031272977197670099525072188924906795726507783915447}{3062851360274591447343732929818426907400340708943550436886023} a^{8} - \frac{456204891789288778047289284873799123115094123055553797555436}{3062851360274591447343732929818426907400340708943550436886023} a^{7} + \frac{998422448260595760372707969906088864070707885920616753021850}{3062851360274591447343732929818426907400340708943550436886023} a^{6} - \frac{1404670977878082354510594775628560093613199570534858759984070}{3062851360274591447343732929818426907400340708943550436886023} a^{5} + \frac{368745531242910701301374554058870005240635440568289930997796}{3062851360274591447343732929818426907400340708943550436886023} a^{4} - \frac{637947895140433379618895211738138987417138734059178961804629}{3062851360274591447343732929818426907400340708943550436886023} a^{3} + \frac{137118354080533410503206274259114458737296358433398789755714}{3062851360274591447343732929818426907400340708943550436886023} a^{2} - \frac{954139173332500597750041631469043906723534511178535917910003}{3062851360274591447343732929818426907400340708943550436886023} a - \frac{1779607993485937710616676623870554561757428503215918644432}{4853964120878908791352984040916682896038574816075357269233}$

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

Class group and class number

$C_{3374}$, which has order $3374$ (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:  \( 2160866793.54 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T926:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1119744
The 267 conjugacy class representatives for t18n926 are not computed
Character table for t18n926 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.4936103895751.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
71Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
28955681Data not computed