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Properties

Label 18.0.58794371317...4544.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 157^{12}$
Root discriminant $58.21$
Ramified primes $2, 157$
Class number $100$ (GRH)
Class group $[10, 10]$ (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![25088, 109312, 238144, 168256, 70016, 48128, 109300, 44248, 9928, 14892, 8529, 12, 200, 356, 266, -20, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 20*x^15 + 266*x^14 + 356*x^13 + 200*x^12 + 12*x^11 + 8529*x^10 + 14892*x^9 + 9928*x^8 + 44248*x^7 + 109300*x^6 + 48128*x^5 + 70016*x^4 + 168256*x^3 + 238144*x^2 + 109312*x + 25088)
 
gp: K = bnfinit(x^18 - 20*x^15 + 266*x^14 + 356*x^13 + 200*x^12 + 12*x^11 + 8529*x^10 + 14892*x^9 + 9928*x^8 + 44248*x^7 + 109300*x^6 + 48128*x^5 + 70016*x^4 + 168256*x^3 + 238144*x^2 + 109312*x + 25088, 1)
 

Normalized defining polynomial

\( x^{18} - 20 x^{15} + 266 x^{14} + 356 x^{13} + 200 x^{12} + 12 x^{11} + 8529 x^{10} + 14892 x^{9} + 9928 x^{8} + 44248 x^{7} + 109300 x^{6} + 48128 x^{5} + 70016 x^{4} + 168256 x^{3} + 238144 x^{2} + 109312 x + 25088 \)

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:  \(-58794371317948797436674598764544=-\,2^{18}\cdot 157^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.21$
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:  $2, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{224} a^{12} + \frac{1}{56} a^{11} - \frac{5}{224} a^{10} - \frac{3}{56} a^{9} + \frac{13}{224} a^{8} - \frac{3}{56} a^{7} - \frac{23}{224} a^{6} - \frac{5}{28} a^{5} - \frac{17}{112} a^{4} + \frac{25}{56} a^{3} + \frac{9}{28} a^{2} + \frac{5}{14} a$, $\frac{1}{448} a^{13} - \frac{1}{448} a^{12} - \frac{11}{448} a^{11} + \frac{13}{448} a^{10} + \frac{3}{448} a^{9} + \frac{1}{64} a^{8} - \frac{5}{448} a^{7} - \frac{37}{448} a^{6} - \frac{25}{112} a^{5} - \frac{19}{224} a^{4} + \frac{3}{28} a^{3} + \frac{1}{8} a^{2} - \frac{1}{7} a$, $\frac{1}{2688} a^{14} - \frac{1}{2688} a^{13} - \frac{1}{2688} a^{12} + \frac{53}{2688} a^{11} - \frac{25}{896} a^{10} + \frac{139}{2688} a^{9} + \frac{13}{2688} a^{8} + \frac{235}{2688} a^{7} + \frac{17}{1344} a^{6} - \frac{59}{448} a^{5} - \frac{73}{672} a^{4} + \frac{97}{336} a^{3} - \frac{5}{56} a^{2} - \frac{2}{7} a + \frac{1}{3}$, $\frac{1}{13440} a^{15} - \frac{1}{1680} a^{13} - \frac{1}{6720} a^{12} + \frac{1}{96} a^{11} + \frac{59}{6720} a^{10} + \frac{1}{960} a^{9} + \frac{7}{960} a^{8} - \frac{1669}{13440} a^{7} + \frac{145}{1344} a^{6} + \frac{1597}{6720} a^{5} + \frac{23}{420} a^{4} - \frac{89}{1680} a^{3} + \frac{67}{140} a^{2} - \frac{5}{42} a + \frac{4}{15}$, $\frac{1}{335220480} a^{16} - \frac{1321}{67044096} a^{15} - \frac{17473}{335220480} a^{14} + \frac{37393}{335220480} a^{13} - \frac{941}{9577728} a^{12} + \frac{6326903}{335220480} a^{11} - \frac{106237}{111740160} a^{10} + \frac{17576423}{335220480} a^{9} - \frac{1596437}{167610240} a^{8} - \frac{701069}{33522048} a^{7} + \frac{111527}{10475640} a^{6} + \frac{1158047}{6983760} a^{5} + \frac{1976131}{41902560} a^{4} + \frac{500543}{1496520} a^{3} - \frac{499267}{2095128} a^{2} + \frac{97799}{374130} a - \frac{367}{37413}$, $\frac{1}{3145545181129638189611520} a^{17} - \frac{1691755621848317}{1572772590564819094805760} a^{16} - \frac{885041593020888245}{26212876509413651580096} a^{15} + \frac{2751320846313752323}{26212876509413651580096} a^{14} - \frac{251323945642119557531}{524257530188273031601920} a^{13} - \frac{248138287251236076887}{786386295282409547402880} a^{12} + \frac{7635172558086390844523}{786386295282409547402880} a^{11} - \frac{1093330934987287926743}{393193147641204773701440} a^{10} + \frac{167670405283790560590781}{3145545181129638189611520} a^{9} + \frac{3662754764718174617137}{104851506037654606320384} a^{8} - \frac{60009461854792789423477}{786386295282409547402880} a^{7} + \frac{109282178653400441357}{1404261241575731334648} a^{6} - \frac{44572576520933969460691}{786386295282409547402880} a^{5} - \frac{3634642709354622575153}{131064382547068257900480} a^{4} - \frac{1241752294555028375171}{21844063757844709650080} a^{3} + \frac{440571907773162163193}{6553219127353412895024} a^{2} - \frac{163156543993277231189}{877663275984832084155} a - \frac{130669268719589557859}{1755326551969664168310}$

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

Class group and class number

$C_{10}\times C_{10}$, which has order $100$ (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:  \( \frac{2201470732769613}{84076261757400855040} a^{17} - \frac{8482691530059}{1050953271967510688} a^{16} + \frac{349031855844513}{42038130878700427520} a^{15} - \frac{3203860008417139}{6005447268385775360} a^{14} + \frac{14971369520955769}{2101906543935021376} a^{13} + \frac{294648747829334687}{42038130878700427520} a^{12} + \frac{199621071053489821}{42038130878700427520} a^{11} - \frac{34853291280162023}{42038130878700427520} a^{10} + \frac{2647302914937761609}{12010894536771550720} a^{9} + \frac{2677805400578093817}{8407626175740085504} a^{8} + \frac{4281306661882259133}{21019065439350213760} a^{7} + \frac{11826700985041035731}{10509532719675106880} a^{6} + \frac{50922915874997382011}{21019065439350213760} a^{5} + \frac{6485724335386073587}{10509532719675106880} a^{4} + \frac{1813745469067572087}{1050953271967510688} a^{3} + \frac{8601662502598132441}{2627383179918776720} a^{2} + \frac{335952868571793955}{65684579497969418} a + \frac{13066109151154473}{9383511356852774} \) (order $4$)
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:  \( 1387709278.61 \) (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

\(\Q(\sqrt{-1}) \), 3.1.98596.1 x3, 3.3.24649.1, 6.0.38884684864.1, 6.0.38884684864.3, 6.0.1577536.1 x2, 9.3.958468597212736.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.1577536.1
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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$157$157.9.6.1$x^{9} + 7065 x^{6} + 16613426 x^{3} + 13060888875$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
157.9.6.1$x^{9} + 7065 x^{6} + 16613426 x^{3} + 13060888875$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$