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Properties

Label 18.0.25815178338...6864.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{44}$
Root discriminant $29.33$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_9\times S_3$ (as 18T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53, -36, 81, -420, 0, 108, 414, 126, 0, -4, 207, 0, 0, 0, 27, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 27*x^14 + 207*x^10 - 4*x^9 + 126*x^7 + 414*x^6 + 108*x^5 - 420*x^3 + 81*x^2 - 36*x + 53)
 
gp: K = bnfinit(x^18 + 27*x^14 + 207*x^10 - 4*x^9 + 126*x^7 + 414*x^6 + 108*x^5 - 420*x^3 + 81*x^2 - 36*x + 53, 1)
 

Normalized defining polynomial

\( x^{18} + 27 x^{14} + 207 x^{10} - 4 x^{9} + 126 x^{7} + 414 x^{6} + 108 x^{5} - 420 x^{3} + 81 x^{2} - 36 x + 53 \)

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:  \(-258151783382020583032356864=-\,2^{18}\cdot 3^{44}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.33$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{2}{17} a^{13} - \frac{1}{17} a^{12} + \frac{6}{17} a^{11} - \frac{4}{17} a^{10} - \frac{3}{17} a^{9} + \frac{2}{17} a^{8} + \frac{4}{17} a^{7} - \frac{6}{17} a^{6} - \frac{3}{17} a^{5} - \frac{4}{17} a^{4} - \frac{5}{17} a^{3} + \frac{7}{17} a^{2} + \frac{4}{17} a - \frac{2}{17}$, $\frac{1}{17} a^{15} - \frac{5}{17} a^{13} + \frac{8}{17} a^{12} + \frac{1}{17} a^{11} + \frac{5}{17} a^{10} + \frac{8}{17} a^{9} + \frac{3}{17} a^{7} - \frac{8}{17} a^{6} + \frac{2}{17} a^{5} + \frac{3}{17} a^{4} + \frac{7}{17} a^{2} + \frac{7}{17} a + \frac{4}{17}$, $\frac{1}{901} a^{16} + \frac{7}{901} a^{15} + \frac{25}{901} a^{14} - \frac{205}{901} a^{13} + \frac{112}{901} a^{12} + \frac{192}{901} a^{11} + \frac{246}{901} a^{10} - \frac{17}{53} a^{9} - \frac{141}{901} a^{8} - \frac{309}{901} a^{7} + \frac{55}{901} a^{6} - \frac{447}{901} a^{5} + \frac{258}{901} a^{4} - \frac{449}{901} a^{3} - \frac{431}{901} a^{2} + \frac{343}{901} a + \frac{1}{17}$, $\frac{1}{635284611710525863} a^{17} - \frac{141386059046440}{635284611710525863} a^{16} + \frac{10881961980641360}{635284611710525863} a^{15} + \frac{4857616205820440}{635284611710525863} a^{14} - \frac{151280167237227152}{635284611710525863} a^{13} + \frac{58580325346194888}{635284611710525863} a^{12} - \frac{185721608246320548}{635284611710525863} a^{11} + \frac{143195504571909386}{635284611710525863} a^{10} - \frac{100207620574289685}{635284611710525863} a^{9} + \frac{15582212373226479}{635284611710525863} a^{8} - \frac{10207437251099547}{635284611710525863} a^{7} + \frac{170148947999835568}{635284611710525863} a^{6} - \frac{179358555263269974}{635284611710525863} a^{5} + \frac{229948336474797666}{635284611710525863} a^{4} + \frac{6047943882442637}{635284611710525863} a^{3} + \frac{286423400023955873}{635284611710525863} a^{2} - \frac{87534211158811483}{635284611710525863} a - \frac{4528187657932765}{11986502107745771}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1392129849823263}{37369683041795639} a^{17} - \frac{607185296487675}{37369683041795639} a^{16} - \frac{400021327070577}{37369683041795639} a^{15} - \frac{353558510578890}{37369683041795639} a^{14} - \frac{37736195194655964}{37369683041795639} a^{13} - \frac{16494113439107079}{37369683041795639} a^{12} - \frac{10889765889975468}{37369683041795639} a^{11} - \frac{9641840493252681}{37369683041795639} a^{10} - \frac{292208864248792427}{37369683041795639} a^{9} - \frac{122844392437665240}{37369683041795639} a^{8} - \frac{82806115356772902}{37369683041795639} a^{7} - \frac{249783981729833358}{37369683041795639} a^{6} - \frac{682842741994370016}{37369683041795639} a^{5} - \frac{472870893531147408}{37369683041795639} a^{4} - \frac{294397377777052872}{37369683041795639} a^{3} + \frac{346609945907324892}{37369683041795639} a^{2} + \frac{25459906136050431}{37369683041795639} a + \frac{1801157999863805}{705088359279163} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1803849.29799 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_9\times S_3$ (as 18T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_9\times S_3$
Character table for $C_9\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), 6.0.419904.1

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

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
2Data not computed
3Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_3e2.6t1.2c1$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_3e2.6t1.2c2$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e3.18t1.1c1$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.2e2_3e3.18t1.1c2$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.9t1.1c1$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.2e2_3e3.18t1.1c3$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.9t1.1c2$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.2e2_3e3.18t1.1c4$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.2e2_3e3.18t1.1c5$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.2e2_3e3.18t1.1c6$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.9t1.1c3$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.9t1.1c4$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.9t1.1c5$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.9t1.1c6$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
2.2e2_3e4.3t2.1c1$2$ $ 2^{2} \cdot 3^{4}$ $x^{3} - 3 x - 4$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3e4.6t5.2c1$2$ $ 2^{2} \cdot 3^{4}$ $x^{6} - 2 x^{3} + 9 x^{2} - 12 x + 5$ $S_3\times C_3$ (as 6T5) $0$ $0$
2.2e2_3e4.6t5.2c2$2$ $ 2^{2} \cdot 3^{4}$ $x^{6} - 2 x^{3} + 9 x^{2} - 12 x + 5$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3e6.18t16.5c1$2$ $ 2^{2} \cdot 3^{6}$ $x^{18} + 27 x^{14} + 207 x^{10} - 4 x^{9} + 126 x^{7} + 414 x^{6} + 108 x^{5} - 420 x^{3} + 81 x^{2} - 36 x + 53$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.2e2_3e6.18t16.5c2$2$ $ 2^{2} \cdot 3^{6}$ $x^{18} + 27 x^{14} + 207 x^{10} - 4 x^{9} + 126 x^{7} + 414 x^{6} + 108 x^{5} - 420 x^{3} + 81 x^{2} - 36 x + 53$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.2e2_3e6.18t16.5c3$2$ $ 2^{2} \cdot 3^{6}$ $x^{18} + 27 x^{14} + 207 x^{10} - 4 x^{9} + 126 x^{7} + 414 x^{6} + 108 x^{5} - 420 x^{3} + 81 x^{2} - 36 x + 53$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.2e2_3e6.18t16.5c4$2$ $ 2^{2} \cdot 3^{6}$ $x^{18} + 27 x^{14} + 207 x^{10} - 4 x^{9} + 126 x^{7} + 414 x^{6} + 108 x^{5} - 420 x^{3} + 81 x^{2} - 36 x + 53$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.2e2_3e6.18t16.5c5$2$ $ 2^{2} \cdot 3^{6}$ $x^{18} + 27 x^{14} + 207 x^{10} - 4 x^{9} + 126 x^{7} + 414 x^{6} + 108 x^{5} - 420 x^{3} + 81 x^{2} - 36 x + 53$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.2e2_3e6.18t16.5c6$2$ $ 2^{2} \cdot 3^{6}$ $x^{18} + 27 x^{14} + 207 x^{10} - 4 x^{9} + 126 x^{7} + 414 x^{6} + 108 x^{5} - 420 x^{3} + 81 x^{2} - 36 x + 53$ $C_9\times S_3$ (as 18T16) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.