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Properties

Label 18.0.99822059247...9728.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 13^{12}$
Root discriminant $31.62$
Ramified primes $2, 3, 13$
Class number $3$ (GRH)
Class group $[3]$ (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![625, -1875, 5775, -7750, 17736, -29334, 33250, -25830, 14373, -5063, 1083, -510, 652, -540, 324, -144, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 144*x^15 + 324*x^14 - 540*x^13 + 652*x^12 - 510*x^11 + 1083*x^10 - 5063*x^9 + 14373*x^8 - 25830*x^7 + 33250*x^6 - 29334*x^5 + 17736*x^4 - 7750*x^3 + 5775*x^2 - 1875*x + 625)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 144*x^15 + 324*x^14 - 540*x^13 + 652*x^12 - 510*x^11 + 1083*x^10 - 5063*x^9 + 14373*x^8 - 25830*x^7 + 33250*x^6 - 29334*x^5 + 17736*x^4 - 7750*x^3 + 5775*x^2 - 1875*x + 625, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 144 x^{15} + 324 x^{14} - 540 x^{13} + 652 x^{12} - 510 x^{11} + 1083 x^{10} - 5063 x^{9} + 14373 x^{8} - 25830 x^{7} + 33250 x^{6} - 29334 x^{5} + 17736 x^{4} - 7750 x^{3} + 5775 x^{2} - 1875 x + 625 \)

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:  \(-998220592474975337089609728=-\,2^{12}\cdot 3^{21}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.62$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{30} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{15} a^{3} - \frac{1}{5} a^{2} + \frac{1}{10} a + \frac{1}{6}$, $\frac{1}{30} a^{10} - \frac{1}{10} a^{7} - \frac{13}{30} a^{4} - \frac{1}{30} a$, $\frac{1}{30} a^{11} - \frac{1}{10} a^{8} - \frac{13}{30} a^{5} - \frac{1}{30} a^{2}$, $\frac{1}{60} a^{12} - \frac{1}{60} a^{10} + \frac{1}{10} a^{8} - \frac{13}{60} a^{6} - \frac{1}{30} a^{4} + \frac{1}{3} a^{3} + \frac{9}{20} a^{2} - \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{60} a^{13} - \frac{1}{60} a^{11} - \frac{1}{5} a^{8} - \frac{7}{60} a^{7} - \frac{1}{30} a^{5} - \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{7}{30} a^{2} + \frac{9}{20} a - \frac{1}{2}$, $\frac{1}{300} a^{14} - \frac{1}{300} a^{13} - \frac{1}{300} a^{12} + \frac{1}{300} a^{11} + \frac{1}{150} a^{9} - \frac{37}{300} a^{8} + \frac{13}{300} a^{7} - \frac{1}{150} a^{6} + \frac{13}{75} a^{5} - \frac{5}{12} a^{4} - \frac{121}{300} a^{3} + \frac{47}{300} a^{2} + \frac{7}{20} a - \frac{1}{6}$, $\frac{1}{119700} a^{15} - \frac{1}{13300} a^{14} + \frac{577}{119700} a^{13} + \frac{28}{4275} a^{12} + \frac{641}{59850} a^{11} + \frac{1187}{119700} a^{10} - \frac{611}{39900} a^{9} + \frac{7193}{39900} a^{8} - \frac{307}{4275} a^{7} - \frac{2357}{119700} a^{6} - \frac{5011}{119700} a^{5} + \frac{169}{5700} a^{4} - \frac{2629}{23940} a^{3} - \frac{1829}{8550} a^{2} + \frac{719}{2394} a + \frac{629}{4788}$, $\frac{1}{119700} a^{16} + \frac{97}{119700} a^{14} + \frac{391}{119700} a^{13} + \frac{757}{119700} a^{12} - \frac{1639}{119700} a^{11} + \frac{29}{3990} a^{10} + \frac{7}{2850} a^{9} - \frac{778}{29925} a^{8} + \frac{28807}{119700} a^{7} + \frac{4616}{29925} a^{6} - \frac{5063}{19950} a^{5} - \frac{7139}{119700} a^{4} + \frac{6113}{119700} a^{3} + \frac{58063}{119700} a^{2} + \frac{5}{4788} a - \frac{169}{532}$, $\frac{1}{1096687025563500} a^{17} - \frac{762221147}{548343512781750} a^{16} - \frac{729518429}{219337405112700} a^{15} + \frac{570634731367}{365562341854500} a^{14} - \frac{2992745943371}{1096687025563500} a^{13} - \frac{26677511239}{4386748102254} a^{12} - \frac{4159763574577}{274171756390875} a^{11} + \frac{3254897744137}{219337405112700} a^{10} + \frac{1958366525387}{274171756390875} a^{9} - \frac{241111472122363}{1096687025563500} a^{8} - \frac{20002641390407}{182781170927250} a^{7} - \frac{109907053265}{461762958132} a^{6} + \frac{1175306599807}{3481546112900} a^{5} + \frac{526876662208321}{1096687025563500} a^{4} + \frac{6387073907017}{365562341854500} a^{3} + \frac{1326992086577}{5222319169350} a^{2} + \frac{10372641727307}{43867481022540} a + \frac{1063882492417}{8773496204508}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{36491442}{45809817275} a^{17} + \frac{126131796251}{18278117092725} a^{16} - \frac{614874759544}{18278117092725} a^{15} + \frac{271929247088}{2611159584675} a^{14} - \frac{4125822315196}{18278117092725} a^{13} + \frac{6558873767312}{18278117092725} a^{12} - \frac{1461387243896}{3655623418545} a^{11} + \frac{677615541916}{2611159584675} a^{10} - \frac{13465040477906}{18278117092725} a^{9} + \frac{13618200857606}{3655623418545} a^{8} - \frac{187580651001592}{18278117092725} a^{7} + \frac{318988733619524}{18278117092725} a^{6} - \frac{381990284139028}{18278117092725} a^{5} + \frac{58078882703384}{3655623418545} a^{4} - \frac{128908900203536}{18278117092725} a^{3} + \frac{18521069199796}{18278117092725} a^{2} - \frac{80694717886}{34815461129} a + \frac{206504804825}{243708227903} \) (order $6$)
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:  \( 1153844.97919 \) (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

\(\Q(\sqrt{-3}) \), 3.1.18252.1 x3, 3.3.169.1, 6.0.999406512.1, 6.0.5913648.2 x2, 6.0.771147.1, 9.3.6080389219008.3 x3

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

Sibling fields

Degree 6 sibling: 6.0.5913648.2
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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$