↑

Properties

Label 18.0.85493948743...2347.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 7^{12}$
Root discriminant $24.27$
Ramified primes $3, 7$
Class number $3$
Class group $[3]$
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3249, -11799, 18306, -14814, 6561, -5184, 11169, -13725, 8532, -1527, -2241, 2565, -1317, 243, 117, -102, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 102*x^15 + 117*x^14 + 243*x^13 - 1317*x^12 + 2565*x^11 - 2241*x^10 - 1527*x^9 + 8532*x^8 - 13725*x^7 + 11169*x^6 - 5184*x^5 + 6561*x^4 - 14814*x^3 + 18306*x^2 - 11799*x + 3249)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 102*x^15 + 117*x^14 + 243*x^13 - 1317*x^12 + 2565*x^11 - 2241*x^10 - 1527*x^9 + 8532*x^8 - 13725*x^7 + 11169*x^6 - 5184*x^5 + 6561*x^4 - 14814*x^3 + 18306*x^2 - 11799*x + 3249, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 102 x^{15} + 117 x^{14} + 243 x^{13} - 1317 x^{12} + 2565 x^{11} - 2241 x^{10} - 1527 x^{9} + 8532 x^{8} - 13725 x^{7} + 11169 x^{6} - 5184 x^{5} + 6561 x^{4} - 14814 x^{3} + 18306 x^{2} - 11799 x + 3249 \)

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:  \(-8549394874383196572862347=-\,3^{31}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.27$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{9} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{684} a^{14} - \frac{1}{36} a^{13} + \frac{1}{684} a^{12} - \frac{11}{342} a^{11} - \frac{2}{57} a^{10} - \frac{3}{38} a^{9} + \frac{1}{19} a^{8} + \frac{53}{228} a^{7} + \frac{67}{228} a^{6} + \frac{91}{228} a^{5} + \frac{4}{57} a^{4} - \frac{4}{57} a^{3} - \frac{17}{57} a^{2} - \frac{15}{76} a + \frac{1}{4}$, $\frac{1}{1368} a^{15} + \frac{5}{342} a^{13} + \frac{73}{1368} a^{12} - \frac{31}{684} a^{11} + \frac{29}{228} a^{10} + \frac{25}{228} a^{9} + \frac{205}{456} a^{8} - \frac{71}{228} a^{7} - \frac{13}{38} a^{6} + \frac{73}{456} a^{5} - \frac{23}{114} a^{4} + \frac{1}{57} a^{3} - \frac{121}{456} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{2736} a^{16} - \frac{1}{2736} a^{15} - \frac{23}{2736} a^{13} - \frac{1}{912} a^{12} + \frac{17}{684} a^{11} - \frac{1}{6} a^{10} - \frac{31}{304} a^{9} - \frac{283}{912} a^{8} - \frac{27}{152} a^{7} - \frac{47}{912} a^{6} - \frac{313}{912} a^{5} - \frac{55}{228} a^{4} + \frac{13}{304} a^{3} + \frac{151}{912} a^{2} - \frac{137}{304} a - \frac{3}{16}$, $\frac{1}{27389019805344} a^{17} - \frac{392072843}{13694509902672} a^{16} - \frac{5833112123}{27389019805344} a^{15} - \frac{18733730959}{27389019805344} a^{14} - \frac{91703257417}{3423627475668} a^{13} + \frac{472846705787}{27389019805344} a^{12} - \frac{31473323417}{2282418317112} a^{11} + \frac{401982540265}{3043224422816} a^{10} + \frac{749151364339}{4564836634224} a^{9} + \frac{4107262706629}{9129673268448} a^{8} - \frac{1461395756653}{9129673268448} a^{7} + \frac{628061644445}{4564836634224} a^{6} - \frac{111978097799}{9129673268448} a^{5} + \frac{1231758155891}{9129673268448} a^{4} - \frac{211530893507}{2282418317112} a^{3} + \frac{317719472947}{1521612211408} a^{2} - \frac{13816242753}{760806105704} a - \frac{44302321081}{160169706464}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{227408632249}{27389019805344} a^{17} + \frac{66961783627}{1141209158556} a^{16} - \frac{1856361128641}{9129673268448} a^{15} + \frac{3766377867181}{9129673268448} a^{14} - \frac{650732735563}{13694509902672} a^{13} - \frac{63640935338333}{27389019805344} a^{12} + \frac{4771536787567}{760806105704} a^{11} - \frac{22757010960057}{3043224422816} a^{10} + \frac{292285700021}{380403052852} a^{9} + \frac{155124436791241}{9129673268448} a^{8} - \frac{329488482150119}{9129673268448} a^{7} + \frac{37435709013611}{1141209158556} a^{6} - \frac{36510862387373}{3043224422816} a^{5} + \frac{92667622137917}{9129673268448} a^{4} - \frac{152427688335587}{4564836634224} a^{3} + \frac{9467240272045}{190201526426} a^{2} - \frac{53386380343923}{1521612211408} a + \frac{1449466872047}{160169706464} \) (order $18$)
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:  \( 636348.279995 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1323.1 x3, \(\Q(\zeta_{9})^+\), 6.0.5250987.1, 6.0.47258883.3 x2, \(\Q(\zeta_{9})\), 9.3.1688134559643.4 x3

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

Sibling fields

Degree 6 sibling: 6.0.47258883.3
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.3e2.6t1.1c1$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.6t1.1c2$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.3e3_7e2.3t2.1c1$2$ $ 3^{3} \cdot 7^{2}$ $x^{3} - 7$ $S_3$ (as 3T2) $1$ $0$
*2 2.3e4_7e2.6t5.9c1$2$ $ 3^{4} \cdot 7^{2}$ $x^{18} - 9 x^{17} + 39 x^{16} - 102 x^{15} + 117 x^{14} + 243 x^{13} - 1317 x^{12} + 2565 x^{11} - 2241 x^{10} - 1527 x^{9} + 8532 x^{8} - 13725 x^{7} + 11169 x^{6} - 5184 x^{5} + 6561 x^{4} - 14814 x^{3} + 18306 x^{2} - 11799 x + 3249$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3e4_7e2.6t5.9c2$2$ $ 3^{4} \cdot 7^{2}$ $x^{18} - 9 x^{17} + 39 x^{16} - 102 x^{15} + 117 x^{14} + 243 x^{13} - 1317 x^{12} + 2565 x^{11} - 2241 x^{10} - 1527 x^{9} + 8532 x^{8} - 13725 x^{7} + 11169 x^{6} - 5184 x^{5} + 6561 x^{4} - 14814 x^{3} + 18306 x^{2} - 11799 x + 3249$ $S_3 \times C_3$ (as 18T3) $0$ $0$

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 *.