↑

Properties

Label 18.0.14488407928...6875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{15}\cdot 7^{15}$
Root discriminant $19.35$
Ramified primes $5, 7$
Class number $2$
Class group $[2]$
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -11, 40, -51, 50, -56, 146, -60, 111, -220, 155, -15, 50, -75, 25, -1, 6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 6*x^16 - x^15 + 25*x^14 - 75*x^13 + 50*x^12 - 15*x^11 + 155*x^10 - 220*x^9 + 111*x^8 - 60*x^7 + 146*x^6 - 56*x^5 + 50*x^4 - 51*x^3 + 40*x^2 - 11*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 + 6*x^16 - x^15 + 25*x^14 - 75*x^13 + 50*x^12 - 15*x^11 + 155*x^10 - 220*x^9 + 111*x^8 - 60*x^7 + 146*x^6 - 56*x^5 + 50*x^4 - 51*x^3 + 40*x^2 - 11*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 6 x^{16} - x^{15} + 25 x^{14} - 75 x^{13} + 50 x^{12} - 15 x^{11} + 155 x^{10} - 220 x^{9} + 111 x^{8} - 60 x^{7} + 146 x^{6} - 56 x^{5} + 50 x^{4} - 51 x^{3} + 40 x^{2} - 11 x + 1 \)

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:  \(-144884079282928466796875=-\,5^{15}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.35$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3107} a^{16} - \frac{1029}{3107} a^{15} - \frac{619}{3107} a^{14} + \frac{290}{3107} a^{13} + \frac{686}{3107} a^{12} + \frac{209}{3107} a^{11} - \frac{794}{3107} a^{10} + \frac{44}{239} a^{9} + \frac{1086}{3107} a^{8} + \frac{217}{3107} a^{7} + \frac{641}{3107} a^{6} - \frac{1472}{3107} a^{5} - \frac{77}{3107} a^{4} - \frac{406}{3107} a^{3} - \frac{630}{3107} a^{2} - \frac{1358}{3107} a + \frac{252}{3107}$, $\frac{1}{18927621722113} a^{17} - \frac{1527611716}{18927621722113} a^{16} - \frac{2416334585455}{18927621722113} a^{15} + \frac{230604422287}{18927621722113} a^{14} + \frac{6600015749889}{18927621722113} a^{13} - \frac{1536541397258}{18927621722113} a^{12} + \frac{5258117002717}{18927621722113} a^{11} - \frac{315342118983}{18927621722113} a^{10} - \frac{8479529438568}{18927621722113} a^{9} - \frac{364015870392}{1455970901701} a^{8} + \frac{829441538911}{18927621722113} a^{7} + \frac{7253862509468}{18927621722113} a^{6} - \frac{7571179474895}{18927621722113} a^{5} + \frac{6426159815769}{18927621722113} a^{4} + \frac{4337294645095}{18927621722113} a^{3} + \frac{1381081673164}{18927621722113} a^{2} + \frac{4614106419386}{18927621722113} a - \frac{6275728452681}{18927621722113}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 7602.871760148733 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\zeta_{7})^+\), 3.1.175.1, 6.0.2100875.1, 6.0.1071875.1, 9.3.12867859375.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
5Data not computed
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.5_7.2t1.1c1$1$ $ 5 \cdot 7 $ $x^{2} - x + 9$ $C_2$ (as 2T1) $1$ $-1$
* 1.5_7.6t1.2c1$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.7.6t1.1c1$1$ $ 7 $ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
1.5_7.6t1.1c1$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
1.7.6t1.1c2$1$ $ 7 $ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.5_7.6t1.2c2$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$ $C_6$ (as 6T1) $0$ $-1$
1.5_7.6t1.1c2$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 2.5e2_7.3t2.1c1$2$ $ 5^{2} \cdot 7 $ $x^{3} - x^{2} + 2 x - 3$ $S_3$ (as 3T2) $1$ $0$
* 2.5e2_7.6t3.1c1$2$ $ 5^{2} \cdot 7 $ $x^{6} - x^{5} + 4 x + 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.5e2_7e2.12t18.1c1$2$ $ 5^{2} \cdot 7^{2}$ $x^{18} - 5 x^{17} + 6 x^{16} - x^{15} + 25 x^{14} - 75 x^{13} + 50 x^{12} - 15 x^{11} + 155 x^{10} - 220 x^{9} + 111 x^{8} - 60 x^{7} + 146 x^{6} - 56 x^{5} + 50 x^{4} - 51 x^{3} + 40 x^{2} - 11 x + 1$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.5e2_7e2.12t18.1c2$2$ $ 5^{2} \cdot 7^{2}$ $x^{18} - 5 x^{17} + 6 x^{16} - x^{15} + 25 x^{14} - 75 x^{13} + 50 x^{12} - 15 x^{11} + 155 x^{10} - 220 x^{9} + 111 x^{8} - 60 x^{7} + 146 x^{6} - 56 x^{5} + 50 x^{4} - 51 x^{3} + 40 x^{2} - 11 x + 1$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.5e2_7e2.6t5.1c1$2$ $ 5^{2} \cdot 7^{2}$ $x^{6} - x^{5} + x^{4} + 6 x^{3} + 8 x^{2} - 29 x + 22$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.5e2_7e2.6t5.1c2$2$ $ 5^{2} \cdot 7^{2}$ $x^{6} - x^{5} + x^{4} + 6 x^{3} + 8 x^{2} - 29 x + 22$ $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.