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Properties

Label 18.0.22034137696...8679.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $29.07$
Ramified primes $3, 7, 11$
Class number $12$
Class group $[2, 6]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 343, 1519, 1484, 3997, -9772, 20896, -24026, 20482, -11878, 5024, -1442, 456, -196, 73, -13, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 7*x^16 - 13*x^15 + 73*x^14 - 196*x^13 + 456*x^12 - 1442*x^11 + 5024*x^10 - 11878*x^9 + 20482*x^8 - 24026*x^7 + 20896*x^6 - 9772*x^5 + 3997*x^4 + 1484*x^3 + 1519*x^2 + 343*x + 49)
 
gp: K = bnfinit(x^18 - 4*x^17 + 7*x^16 - 13*x^15 + 73*x^14 - 196*x^13 + 456*x^12 - 1442*x^11 + 5024*x^10 - 11878*x^9 + 20482*x^8 - 24026*x^7 + 20896*x^6 - 9772*x^5 + 3997*x^4 + 1484*x^3 + 1519*x^2 + 343*x + 49, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 7 x^{16} - 13 x^{15} + 73 x^{14} - 196 x^{13} + 456 x^{12} - 1442 x^{11} + 5024 x^{10} - 11878 x^{9} + 20482 x^{8} - 24026 x^{7} + 20896 x^{6} - 9772 x^{5} + 3997 x^{4} + 1484 x^{3} + 1519 x^{2} + 343 x + 49 \)

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:  \(-220341376966031977018118679=-\,3^{9}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.07$
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:  $3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{11} + \frac{3}{20} a^{10} - \frac{3}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{20} a^{6} + \frac{1}{10} a^{5} + \frac{1}{4} a^{4} + \frac{1}{20} a^{3} + \frac{7}{20} a^{2} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{20} a^{14} + \frac{1}{20} a^{12} - \frac{1}{10} a^{11} - \frac{3}{20} a^{10} - \frac{1}{20} a^{9} - \frac{1}{20} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{4} a^{5} - \frac{1}{5} a^{4} - \frac{3}{20} a^{3} - \frac{3}{20} a^{2} - \frac{3}{20} a + \frac{1}{4}$, $\frac{1}{140} a^{15} + \frac{3}{140} a^{14} + \frac{1}{140} a^{12} + \frac{1}{14} a^{11} + \frac{1}{20} a^{10} + \frac{29}{140} a^{9} - \frac{3}{20} a^{8} + \frac{13}{70} a^{7} - \frac{1}{7} a^{6} + \frac{7}{20} a^{5} + \frac{2}{7} a^{4} - \frac{13}{140} a^{3} - \frac{7}{20} a^{2} - \frac{2}{5} a - \frac{1}{20}$, $\frac{1}{64540} a^{16} - \frac{43}{16135} a^{15} + \frac{69}{4610} a^{14} - \frac{79}{6454} a^{13} - \frac{5709}{64540} a^{12} + \frac{91}{4610} a^{11} - \frac{12417}{64540} a^{10} + \frac{753}{9220} a^{9} + \frac{7831}{64540} a^{8} + \frac{1901}{32270} a^{7} + \frac{293}{922} a^{6} - \frac{3677}{64540} a^{5} - \frac{3743}{16135} a^{4} - \frac{4059}{9220} a^{3} - \frac{687}{2305} a^{2} + \frac{379}{9220} a + \frac{1507}{4610}$, $\frac{1}{11903630297992011469100} a^{17} + \frac{44541124973854457}{11903630297992011469100} a^{16} - \frac{41459965694722884371}{11903630297992011469100} a^{15} - \frac{172763659599917428229}{11903630297992011469100} a^{14} - \frac{66956432251312914029}{2975907574498002867275} a^{13} - \frac{191798426264147977573}{2975907574498002867275} a^{12} - \frac{550972640732088292493}{5951815148996005734550} a^{11} - \frac{520213544052571099092}{2975907574498002867275} a^{10} + \frac{784601960338468394813}{5951815148996005734550} a^{9} - \frac{222500716822103092531}{5951815148996005734550} a^{8} - \frac{25243749169197226638}{595181514899600573455} a^{7} + \frac{756593091751333672156}{2975907574498002867275} a^{6} - \frac{138008829764428167726}{595181514899600573455} a^{5} - \frac{1990980667599294854351}{5951815148996005734550} a^{4} + \frac{607058939274725586391}{2380726059598402293820} a^{3} + \frac{366323798735226823187}{1700518613998858781300} a^{2} + \frac{602269550941481521279}{1700518613998858781300} a + \frac{468277201052490271983}{1700518613998858781300}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 81487.9586443 \)
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

\(\Q(\sqrt{-231}) \), 3.1.231.1 x3, \(\Q(\zeta_{7})^+\), 6.0.12326391.1, 6.0.603993159.2 x2, 6.0.603993159.1, 9.3.29595664791.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.603993159.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{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_7_11.2t1.1c1$1$ $ 3 \cdot 7 \cdot 11 $ $x^{2} - x + 58$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3_7_11.6t1.2c1$1$ $ 3 \cdot 7 \cdot 11 $ $x^{6} - x^{5} + 57 x^{4} - 57 x^{3} + 953 x^{2} - 953 x + 4537$ $C_6$ (as 6T1) $0$ $-1$
* 1.3_7_11.6t1.2c2$1$ $ 3 \cdot 7 \cdot 11 $ $x^{6} - x^{5} + 57 x^{4} - 57 x^{3} + 953 x^{2} - 953 x + 4537$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.3_7_11.3t2.1c1$2$ $ 3 \cdot 7 \cdot 11 $ $x^{3} - x^{2} + 3$ $S_3$ (as 3T2) $1$ $0$
*2 2.3_7e2_11.6t5.3c1$2$ $ 3 \cdot 7^{2} \cdot 11 $ $x^{18} - 4 x^{17} + 7 x^{16} - 13 x^{15} + 73 x^{14} - 196 x^{13} + 456 x^{12} - 1442 x^{11} + 5024 x^{10} - 11878 x^{9} + 20482 x^{8} - 24026 x^{7} + 20896 x^{6} - 9772 x^{5} + 3997 x^{4} + 1484 x^{3} + 1519 x^{2} + 343 x + 49$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.3_7e2_11.6t5.3c2$2$ $ 3 \cdot 7^{2} \cdot 11 $ $x^{18} - 4 x^{17} + 7 x^{16} - 13 x^{15} + 73 x^{14} - 196 x^{13} + 456 x^{12} - 1442 x^{11} + 5024 x^{10} - 11878 x^{9} + 20482 x^{8} - 24026 x^{7} + 20896 x^{6} - 9772 x^{5} + 3997 x^{4} + 1484 x^{3} + 1519 x^{2} + 343 x + 49$ $S_3 \times C_3$ (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 *.