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Properties

Label 18.4.23431125972...1344.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 71$
Root discriminant $55.31$
Ramified primes $2, 3, 7, 41, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![118391, -19523, 149163, -84904, 97174, -39513, -35570, 49129, -52371, 22741, -11420, -142, 628, -1099, 458, -155, 42, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 42*x^16 - 155*x^15 + 458*x^14 - 1099*x^13 + 628*x^12 - 142*x^11 - 11420*x^10 + 22741*x^9 - 52371*x^8 + 49129*x^7 - 35570*x^6 - 39513*x^5 + 97174*x^4 - 84904*x^3 + 149163*x^2 - 19523*x + 118391)
 
gp: K = bnfinit(x^18 - 6*x^17 + 42*x^16 - 155*x^15 + 458*x^14 - 1099*x^13 + 628*x^12 - 142*x^11 - 11420*x^10 + 22741*x^9 - 52371*x^8 + 49129*x^7 - 35570*x^6 - 39513*x^5 + 97174*x^4 - 84904*x^3 + 149163*x^2 - 19523*x + 118391, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 42 x^{16} - 155 x^{15} + 458 x^{14} - 1099 x^{13} + 628 x^{12} - 142 x^{11} - 11420 x^{10} + 22741 x^{9} - 52371 x^{8} + 49129 x^{7} - 35570 x^{6} - 39513 x^{5} + 97174 x^{4} - 84904 x^{3} + 149163 x^{2} - 19523 x + 118391 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-23431125972725566736546343481344=-\,2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 71\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.31$
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:  $2, 3, 7, 41, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{3} - \frac{1}{16} a^{2} - \frac{1}{4} a + \frac{5}{16}$, $\frac{1}{32} a^{14} - \frac{1}{16} a^{12} + \frac{3}{32} a^{11} + \frac{1}{32} a^{10} + \frac{3}{32} a^{9} - \frac{1}{16} a^{8} + \frac{3}{32} a^{7} - \frac{7}{32} a^{6} - \frac{3}{16} a^{5} + \frac{1}{32} a^{4} + \frac{1}{4} a^{3} - \frac{5}{32} a^{2} + \frac{5}{32} a - \frac{3}{32}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{32} a^{13} - \frac{3}{64} a^{12} - \frac{1}{32} a^{11} - \frac{3}{32} a^{10} + \frac{3}{64} a^{9} + \frac{5}{64} a^{8} - \frac{5}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{64} a^{5} + \frac{15}{64} a^{4} + \frac{27}{64} a^{3} + \frac{5}{32} a^{2} + \frac{1}{4} a + \frac{11}{64}$, $\frac{1}{256} a^{16} + \frac{1}{256} a^{14} - \frac{5}{256} a^{13} - \frac{13}{256} a^{12} + \frac{5}{64} a^{11} + \frac{17}{256} a^{10} + \frac{1}{64} a^{9} - \frac{29}{256} a^{8} + \frac{51}{256} a^{7} + \frac{5}{64} a^{6} + \frac{11}{128} a^{5} + \frac{23}{128} a^{4} + \frac{37}{256} a^{3} - \frac{61}{128} a^{2} + \frac{63}{256} a + \frac{15}{256}$, $\frac{1}{73933173323494163130254626281238524928} a^{17} - \frac{19138633012379023625296669959730939}{73933173323494163130254626281238524928} a^{16} + \frac{14200266724105528974052068271293905}{73933173323494163130254626281238524928} a^{15} - \frac{26447768065857977308335320877470327}{2310411666359192597820457071288703904} a^{14} - \frac{1093550024836456227732222795866858803}{36966586661747081565127313140619262464} a^{13} + \frac{2433783185940151278180936475415410595}{73933173323494163130254626281238524928} a^{12} - \frac{1344607817050224711235359691249479675}{73933173323494163130254626281238524928} a^{11} + \frac{2835792012498939152911481423194495785}{73933173323494163130254626281238524928} a^{10} - \frac{1135818950556258724134616553787824169}{73933173323494163130254626281238524928} a^{9} - \frac{1188239276468367971137709561833753303}{36966586661747081565127313140619262464} a^{8} + \frac{13543042506539836460015408548782467715}{73933173323494163130254626281238524928} a^{7} - \frac{9181527217526710268468974386634783923}{36966586661747081565127313140619262464} a^{6} - \frac{4275433990534615997705657583385217733}{18483293330873540782563656570309631232} a^{5} + \frac{12581462825437023261742200228698967851}{73933173323494163130254626281238524928} a^{4} - \frac{17426054284188835268366560075438869361}{73933173323494163130254626281238524928} a^{3} + \frac{10901363103746582670786127256932176109}{73933173323494163130254626281238524928} a^{2} + \frac{15500911139801320139625535837276935197}{36966586661747081565127313140619262464} a + \frac{6428480457064435806958318369304795531}{73933173323494163130254626281238524928}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $10$
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:  \( 904544660.09 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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

Sibling fields

Degree 18 siblings: 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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
41Data not computed
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.6.0.1$x^{6} - 2 x + 13$$1$$6$$0$$C_6$$[\ ]^{6}$
71.6.0.1$x^{6} - 2 x + 13$$1$$6$$0$$C_6$$[\ ]^{6}$