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Properties

Label 18.0.15522256482...6171.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{9}\cdot 37^{12}$
Root discriminant $36.83$
Ramified primes $11, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
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![13121, -49979, 58816, -23408, 31004, -38174, 41267, -32168, 19479, -11263, 7548, -5348, 3487, -1820, 748, -236, 55, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 55*x^16 - 236*x^15 + 748*x^14 - 1820*x^13 + 3487*x^12 - 5348*x^11 + 7548*x^10 - 11263*x^9 + 19479*x^8 - 32168*x^7 + 41267*x^6 - 38174*x^5 + 31004*x^4 - 23408*x^3 + 58816*x^2 - 49979*x + 13121)
 
gp: K = bnfinit(x^18 - 9*x^17 + 55*x^16 - 236*x^15 + 748*x^14 - 1820*x^13 + 3487*x^12 - 5348*x^11 + 7548*x^10 - 11263*x^9 + 19479*x^8 - 32168*x^7 + 41267*x^6 - 38174*x^5 + 31004*x^4 - 23408*x^3 + 58816*x^2 - 49979*x + 13121, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 55 x^{16} - 236 x^{15} + 748 x^{14} - 1820 x^{13} + 3487 x^{12} - 5348 x^{11} + 7548 x^{10} - 11263 x^{9} + 19479 x^{8} - 32168 x^{7} + 41267 x^{6} - 38174 x^{5} + 31004 x^{4} - 23408 x^{3} + 58816 x^{2} - 49979 x + 13121 \)

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:  \(-15522256482134329706192486171=-\,11^{9}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.83$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{1023} a^{14} - \frac{7}{1023} a^{13} + \frac{100}{1023} a^{12} - \frac{509}{1023} a^{11} + \frac{35}{1023} a^{10} + \frac{232}{1023} a^{9} + \frac{1}{11} a^{8} + \frac{28}{341} a^{7} + \frac{178}{1023} a^{6} - \frac{386}{1023} a^{5} + \frac{48}{341} a^{4} - \frac{101}{1023} a^{3} + \frac{499}{1023} a^{2} - \frac{11}{31} a - \frac{395}{1023}$, $\frac{1}{1023} a^{15} + \frac{17}{341} a^{13} - \frac{50}{341} a^{12} - \frac{153}{341} a^{11} - \frac{205}{1023} a^{10} - \frac{329}{1023} a^{9} + \frac{394}{1023} a^{8} + \frac{28}{341} a^{7} - \frac{163}{1023} a^{6} - \frac{57}{341} a^{5} - \frac{116}{1023} a^{4} - \frac{208}{1023} a^{3} + \frac{61}{1023} a^{2} - \frac{208}{1023} a - \frac{126}{341}$, $\frac{1}{1855062015262467} a^{16} - \frac{8}{1855062015262467} a^{15} - \frac{15241629005}{168642001387497} a^{14} + \frac{391201811175}{618354005087489} a^{13} + \frac{37921043899120}{1855062015262467} a^{12} - \frac{242783134030909}{1855062015262467} a^{11} + \frac{189281893383923}{1855062015262467} a^{10} - \frac{182662830244039}{618354005087489} a^{9} + \frac{272079121154422}{1855062015262467} a^{8} - \frac{172983393140717}{618354005087489} a^{7} - \frac{84904965618124}{1855062015262467} a^{6} - \frac{848660616275390}{1855062015262467} a^{5} + \frac{684225955065}{19946903389919} a^{4} + \frac{416968083391322}{1855062015262467} a^{3} + \frac{773686179101299}{1855062015262467} a^{2} + \frac{488712103813097}{1855062015262467} a - \frac{12862055384573}{618354005087489}$, $\frac{1}{866313961127572089} a^{17} + \frac{75}{288771320375857363} a^{16} - \frac{194196627259222}{866313961127572089} a^{15} + \frac{147071505363868}{866313961127572089} a^{14} - \frac{132076226688981958}{866313961127572089} a^{13} - \frac{88501452188247125}{866313961127572089} a^{12} - \frac{311280666755592775}{866313961127572089} a^{11} - \frac{28808861080134926}{288771320375857363} a^{10} + \frac{58451913520627839}{288771320375857363} a^{9} - \frac{197953850386729927}{866313961127572089} a^{8} + \frac{32609014194861406}{78755814647961099} a^{7} - \frac{71105596999020731}{288771320375857363} a^{6} + \frac{296512054261752029}{866313961127572089} a^{5} + \frac{119780079087924320}{288771320375857363} a^{4} + \frac{251313496613195548}{866313961127572089} a^{3} - \frac{119045348347566959}{288771320375857363} a^{2} - \frac{196158051495872410}{866313961127572089} a + \frac{43473985929344874}{288771320375857363}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2475749.23259 \) (assuming GRH)
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

\(\Q(\sqrt{-11}) \), 3.1.15059.1 x3, 3.3.1369.1, 6.0.2494508291.1, 6.0.2494508291.2, 6.0.1822139.1 x2, 9.3.3414981850379.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.1822139.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$